(Math) Evalution

Elaborate answer extraction and correctness judgement (for mathematical evaluation). 
from dart_math.eval import *

math_evaluator = EvaluatorMathBatch()
WARNING 10-07 08:27:11 _custom_ops.py:14] Failed to import from vllm._C with ImportError('libcuda.so.1: cannot open shared object file: No such file or directory')

Elaborate Mathematical Evaluation Pipeline

source

EvaluatorMath

 EvaluatorMath (strict_extract:bool=False,
                use_orig_eq_for_olympiadbench:bool=True,
                include_percentage:bool=True, rel_tol:float=1e-09,
                abs_tol:float=1e-08, percent_rel_tol:float=0.001,
                ascii_only:bool=True)

Evaluator for math problems, capable of extracting answer segment from complex resp and processing various mathematical objects (e.g. fractions, symbolic expressions, matrices, vectors) and special text (e.g. bool values).

Type Default Details
strict_extract bool False
use_orig_eq_for_olympiadbench bool True Whether to use the original implementation of eq for OlympiadBench.
For OlympiadBench, by default, we use the official implementation of eq by He et al. (2024),
which utilizing the numerical error range information provided with query,
but keep the extract_nas of ours,
because the official implementation fails to extract a non-negligible part of answers, especially for base model ICL.
You could set use_orig_eq_for_olympiadbench to False to use our implementation of eq
for better consistency across benchmarks in our evaluation setting.
include_percentage bool True Whether to include percentage comparisons.
rel_tol float 1e-09 The relative tolerance for numerical comparisons.
abs_tol float 1e-08 The absolute tolerance for numerical comparisons. Necessary for precision issues.
percent_rel_tol float 0.001 The relative tolerance for percentage comparisons. Relative for different surface forms (e.g. 99% v.s. 0.99).
ascii_only bool True Only allowing ASCII characters

EvaluatorMath implements an elaborate evaluation pipeline for mathematical reasoning tasks.

Accurately Extracting Answer Strings

EvaluatorMath can:

  1. extract short answers from long responses rather accurately
  2. and normalize into a mathematical expression.
# MATH-style boxed answer
math_evaluator.extract_ans("Therefore, $1+1=\\boxed{2}$.")
'2'
# Answer around "answer"
math_evaluator.extract_ans(
    "Both $1$ and $11$ divide $11,$ so $\\boxed{11}=2$, and since $1,$ $2,$ $4,$ $5,$ $10,$ and $20$ divide $20,$ then $\\boxed{20}=6$. The inner expression, $\\boxed{11}\\times\\boxed{20}=2\\times6=12$. Finally, $\\boxed{12}=6$ because $1,$ $2,$ $3,$ $4,$ $6,$ and $12$ divide $12.$\n\nTherefore, $6$ is our answer. Please note that we have not boxed the correct answer as we normally do, as that would be especially confusing for this problem."
)
'6'
# Use the last number by default
math_evaluator.extract_ans(
    'First, we need to count the total number of letters in the word "CIRCLE". There are 6 letters.\n\nNext, we need to count the number of distinct letters. There are 6 distinct letters in the word "CIRCLE": C, I, R, L, E, and G.\n\nNow, let\'s consider the arrangements of the distinct letters. The number of ways to arrange n distinct items is n factorial (n!). So, we have 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways to arrange the distinct letters.\n\nHowever, the word "CIRCLE" has one letter that repeats (the letter \'C\' repeats twice). We have over-counted the number of distinct arrangements by including arrangements that are just rotations of each other (for example, "CIRCLE" and "LCIRCE" are considered different arrangements here, but they are the same word when read).\n\nTo correct for this, we divide the total number of arrangements by the number of ways to arrange the repeated letters. The number of ways to arrange 2 identical items is 2! = 2 × 1 = 2. So, we divide the total number of arrangements by 2 to get the correct number of distinct arrangements.\n\nTherefore, the number of ways to arrange the letters of the word "CIRCLE" is 720 ÷ 2 = 360.'
)
# More cases ...
'360'
# Normalize fraction
math_evaluator.extract_ans("The answer is 1/2")
'\\frac{1}{2}'
# Normalize pmatrix
math_evaluator.extract_ans(
    "The answer is \\begin{pmatrix} 3 \\\\ \\frac{\\pi}{2} \\end{pmatrix}"
)
# More cases ...
'\\begin{array}3\\\\frac{\\pi}{2}\\end{array}'

Correctly Processing Various Mathematical Objects / Special Text

EvaluatorMath, based on regular expressions and SymPy symbolic calculation, is able to correctly process

  • most mathematical objects such as matrices (vectors), intervals, symbols besides numbers,
  • as well as some special texts like bool expressions, dates and times.
math_evaluator.eq("x+y", "y+x") == True  # Expression
True
math_evaluator.eq("\\frac{1}{2}", "0.5") == True  # LaTeX
True
math_evaluator.eq(
    "\\begin{array}1\\\\2\\end{array}",
    "1,2",
)  # Matrix (Vector)
True
math_evaluator.eq("{1,2}", "{2,1}", compare_sets=True)  # Set
True
math_evaluator.eq("no", "false")  # Bool
# More mathematical objects and special texts ...
True

More test cases:

Code 
test_eq(math_evaluator.eq("251,7\\\\ \\noindent", "0"), False)
test_eq(math_evaluator.eq("3.54*10^{-7}", "3.54e-07"), True)
test_eq(math_evaluator.eq(r"\frac{1}{2}", "0.5"), True)
test_eq(math_evaluator.eq("1", "100"), False)
test_eq(math_evaluator.eq("100", "1"), False)
test_eq(math_evaluator.eq("3.04", "0.0304", False), True)
test_eq(math_evaluator.eq(["0.0304", 0.0304], "3.04"), True)
test_eq(math_evaluator.eq("x<-1", "x>3"), False)
test_eq(
    math_evaluator.eq("(-\\infty,0)\\cup(0,\\infty)", "(-\\infty,0)\\cup(0,\\infty)"),
    True,
)
test_eq(math_evaluator.eq("1+2,2+1", "2+1,1+2"), True)
test_eq(math_evaluator.eq(5, 5), True)
test_eq(math_evaluator.eq(0.1 + 0.2, 0.3), True)  # `0.1 + 0.2 == 0.3` is `False`
test_eq(math_evaluator.eq("x + y", "y + x"), True)
test_eq(math_evaluator.eq("C", "C"), True)
test_eq(math_evaluator.eq("1,234", "1234"), True)
test_eq(math_evaluator.eq("12,34", "(12,34)"), True)

test_eq(math_evaluator.eq("\\$ 5", "5"), True)
test_eq(math_evaluator.eq("3 * \\sqrt{13}", "3\\sqrt{13}"), True)
test_eq(math_evaluator.eq("\\pi/2", "\\frac{\\pi}{2}"), True)
test_eq(math_evaluator.eq("(3,\\pi/2)", "(3,\\frac{\\pi}{2})"), True)
test_eq(math_evaluator.eq("23000", "\\$23{,}000"), True)
test_eq(
    math_evaluator.eq(r"\left(1,2\right)", r"\left(2,1\right)", compare_sets=True), True
)
test_eq(math_evaluator.eq("White", "white"), True)
test_eq(math_evaluator.eq("[0,3)", "[0,1]"), False)
test_eq(math_evaluator.eq("[0,1]", "[0,3)"), False)
test_eq(math_evaluator.eq("1001.5", "1001"), False)
test_eq(math_evaluator.eq("\\frac{2003}{2}", "1001"), False)

source

EvaluatorMathBatch

 EvaluatorMathBatch (strict_extract:bool=False,
                     use_orig_eq_for_olympiadbench:bool=True,
                     include_percentage:bool=True, rel_tol:float=1e-09,
                     abs_tol:float=1e-08, percent_rel_tol:float=0.001,
                     ascii_only:bool=True, timeout:int=5)

Batch evaluator for math problems, capable of extracting answer segment from complex resp and processing various mathematical objects (e.g. fractions, symbolic expressions, matrices, vectors) and special text (e.g. bool values).

Type Default Details
strict_extract bool False
use_orig_eq_for_olympiadbench bool True Whether to use the original implementation of eq for OlympiadBench.
For OlympiadBench, by default, we use the official implementation of eq by He et al. (2024),
which utilizing the numerical error range information provided with query,
but keep the extract_nas of ours,
because the official implementation fails to extract a non-negligible part of answers, especially for base model ICL.
You could set use_orig_eq_for_olympiadbench to False to use our implementation of eq
for better consistency across benchmarks in our evaluation setting.
include_percentage bool True Whether to include percentage comparisons.
rel_tol float 1e-09 The relative tolerance for numerical comparisons.
abs_tol float 1e-08 The absolute tolerance for numerical comparisons. Necessary for precision issues.
percent_rel_tol float 0.001 The absolute tolerance for percentage comparisons.
ascii_only bool True Only allowing ASCII characters
timeout int 5

SymPy symbolic calculation causes risks of ex-long evaluation time.

To address this, we implement EvaluatorMathBatch to evaluate in batch with timeout but still efficiently (based on asyncio coroutines instead of multiprocessing in previous implementations).

answers, corrects = math_evalutor.batch_eval(resp_samples)

API Reference

source

EvaluatorBase

 EvaluatorBase (strict_extract:bool=False)

Base class for evaluators.

source

EvaluatorBatchBase

 EvaluatorBatchBase (strict_extract:bool=False, timeout:int=5)

Base class for batch evaluators, providing additional method for batch evaluation.

Type Default Details
strict_extract bool False
timeout int 5 The timeout for each evaluation in seconds.

Parsing LaTeX

Interval

from dart_math.eval import latex2sympy_interval
latex2sympy_interval("(-11,-10)\\cup\\{-\\sqrt{110}\\}")

\(\displaystyle \left(-11, -10\right)\)

latex2sympy_interval("(-\\infty, 0) \\cup (0, \\infty)")

\(\displaystyle \left(-\infty, 0\right) \cup \left(0, \infty\right)\)

latex2sympy_interval("(a+b,b]")

\(\displaystyle \left(a + b, b\right]\)

Matrix / Vector

math_evaluator.latex2matrix(r"\sqrt{400\cos^2(9\pi/44)},\frac{\pi}{4}")

\(\displaystyle \left[\begin{matrix}\sqrt{400 \cos^{2}{\left(\frac{9 \pi}{44} \right)}} & \frac{\pi}{4}\end{matrix}\right]\)

math_evaluator.latex2matrix(
    r"\begin{pmatrix} \frac{1}{2} & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{pmatrix}"
)

\(\displaystyle \left[\begin{matrix}\frac{1}{2} & 0 & - \frac{\sqrt{3}}{2}\\0 & 1 & 0\\\frac{\sqrt{3}}{2} & 0 & \frac{1}{2}\end{matrix}\right]\)

test_eq(
    math_evaluator.latex2matrix("\\begin{pmatrix}-18\\\\-49\\\\96\\end{pmatrix}"),
    Matrix([[-18, -49, 96]]),
)
test_eq(
    math_evaluator.latex2matrix("\\begin{pmatrix} 2 & 3 \\\\ 0 & -2 \\end{pmatrix}"),
    Matrix([[2, 3], [0, -2]]),
)

Normalization

test_eq(math_evaluator.norm_math_str("251,7\\\\ \\noindent"), "251,7")
test_eq(fix_a_slash_b("(3/4)\\sqrt{3}"), "(\\frac{3}{4})\\sqrt{3}")
test_eq(math_evaluator.norm_pm("x\\pmy"), "x-y,x+y")
test_eq(math_evaluator.norm_pm("a\\mpb"), "a-b,a+b")
test_eq(math_evaluator.norm_pm("1\\pm\\sqrt{19}"), "1-\\sqrt{19},1+\\sqrt{19}")
test_eq(math_evaluator.norm_pm(r"\{1\pm\sqrt{5},-2\}"), "1-\\sqrt{5},1+\\sqrt{5},-2")
test_eq(
    math_evaluator.norm_pm("\\(\\frac{1\\pm\\sqrt{17}}{4}\\)"),
    "\\frac{1-\\sqrt{17}}{4},\\frac{1+\\sqrt{17}}{4}",
)
test_eq(
    math_evaluator.norm_pm(r"\frac{1\pm\sqrt{1-\frac{2}{\sqrt{3}}}}{1}"),
    "\\frac{1-\\sqrt{1-\\frac{2}{\\sqrt{3}}}}{1},\\frac{1+\\sqrt{1-\\frac{2}{\\sqrt{3}}}}{1}",
)
test_eq(norm_deg(r"20^\circ"), r"20")
test_eq(norm_deg(r"\sin 20^\circ"), r"\sin {20*\frac{\pi}{180}}")
test_eq(math_evaluator.norm_basic_fn(r"sinx"), r"\sin^{1}x")
test_eq(math_evaluator.norm_basic_fn(r"\sin^2x"), r"\sin^{2}x")

Processing Sets

test_eq(math_evaluator.extract_set("{2,1}"), ["1", "2"])
test_eq(is_set("{2,1}"), True)
test_eq(is_set("orange"), False)
test_eq(is_set("x<-1orx>3"), True)
test_eq(is_set("(3/4)sqrt(3)"), False)

Manipulating Strings

test_eq(math_evaluator.remove_first_paren_pair("{white}", "{"), "white")
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