Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 7.3s

Alternatives: 7

Speedup: 2.9×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

   2. pow1/3 98.8%

      \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

   3. pow3 98.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. clear-num 98.8%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 98.8%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. pow-pow 98.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   7. metadata-eval 98.8%

      \[\leadsto {\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}}} \]

   2. +-commutative 98.3%

      \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{e^{b} + e^{a}}}{e^{a}}\right)}^{-3}} \]

6. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}} \]

7. Taylor expanded in b around inf 98.8%

   \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]
8. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}} \]

   2. rem-exp-log 98.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}} \]

   3. div-exp 98.8%

      \[\leadsto \color{blue}{e^{a - \log \left(e^{b} + e^{a}\right)}} \]

   4. sub-neg 98.8%

      \[\leadsto e^{\color{blue}{a + \left(-\log \left(e^{b} + e^{a}\right)\right)}} \]

   5. +-commutative 98.8%

      \[\leadsto e^{\color{blue}{\left(-\log \left(e^{b} + e^{a}\right)\right) + a}} \]

   6. neg-sub0 98.8%

      \[\leadsto e^{\color{blue}{\left(0 - \log \left(e^{b} + e^{a}\right)\right)} + a} \]

   7. associate-+l- 98.8%

      \[\leadsto e^{\color{blue}{0 - \left(\log \left(e^{b} + e^{a}\right) - a\right)}} \]

   8. metadata-eval 98.8%

      \[\leadsto e^{\color{blue}{\log 1} - \left(\log \left(e^{b} + e^{a}\right) - a\right)} \]

   9. rem-log-exp 98.8%

      \[\leadsto e^{\log 1 - \left(\log \left(e^{b} + e^{a}\right) - \color{blue}{\log \left(e^{a}\right)}\right)} \]

   10. log-div 98.8%

       \[\leadsto e^{\log 1 - \color{blue}{\log \left(\frac{e^{b} + e^{a}}{e^{a}}\right)}} \]

   11. log-div 98.8%

       \[\leadsto e^{\color{blue}{\log \left(\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}\right)}} \]

   12. unpow-1 98.8%

       \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)}} \]

   13. rem-exp-log 98.8%

       \[\leadsto \color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}} \]

   14. unpow-1 98.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}} \]

   15. rem-exp-log 98.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}}{e^{a}}} \]

   16. exp-diff 98.8%

       \[\leadsto \frac{1}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right) - a}}} \]

9. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{\left(-a\right) + b}}} \]

10. Final simplification 100.0%

    \[\leadsto \frac{1}{1 + e^{b - a}} \]
11. Add Preprocessing

Alternative 2: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.0001:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;e^{b} \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.0001)
   (+ 1.0 (exp b))
   (if (<= (exp b) 2e+19)
     (/ 1.0 (- 2.0 a))
     (* -0.020833333333333332 (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.0001) {
		tmp = 1.0 + exp(b);
	} else if (exp(b) <= 2e+19) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -0.020833333333333332 * pow(a, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 0.0001d0) then
        tmp = 1.0d0 + exp(b)
    else if (exp(b) <= 2d+19) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-0.020833333333333332d0) * (a ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 0.0001) {
		tmp = 1.0 + Math.exp(b);
	} else if (Math.exp(b) <= 2e+19) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(b) <= 0.0001:
		tmp = 1.0 + math.exp(b)
	elif math.exp(b) <= 2e+19:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.0001)
		tmp = Float64(1.0 + exp(b));
	elseif (exp(b) <= 2e+19)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 0.0001)
		tmp = 1.0 + exp(b);
	elseif (exp(b) <= 2e+19)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -0.020833333333333332 * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.0001], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 2e+19], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 1.00000000000000005e-4

   1. Initial program 96.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 96.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 96.5%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 96.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot 0.3333333333333333}} \]

      4. pow3 96.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 96.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 0.3333333333333333} \]

      6. log-div 96.5%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 96.6%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333} \]

   4. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333}} \]

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(1 + e^{b}\right)}} \]
   6. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\color{blue}{-\log \left(1 + e^{b}\right)}} \]

      2. log1p-def 100.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   7. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 98.3%

         \[\leadsto e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. sqrt-unprod 98.9%

         \[\leadsto e^{\color{blue}{\sqrt{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}}} \]

      3. sqr-neg 98.9%

         \[\leadsto e^{\sqrt{\color{blue}{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}}} \]

      4. sqrt-unprod 98.9%

         \[\leadsto e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}}} \]

      5. add-sqr-sqrt 98.9%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

      6. log1p-udef 98.9%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{b}\right)}} \]

      7. rem-exp-log 98.9%

         \[\leadsto \color{blue}{1 + e^{b}} \]

      8. +-commutative 98.9%

         \[\leadsto \color{blue}{e^{b} + 1} \]

   9. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{e^{b} + 1} \]

   if 1.00000000000000005e-4 < (exp.f64 b) < 2e19

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 98.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 99.2%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow3 99.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 99.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 99.2%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 99.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 99.2%

         \[\leadsto {\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 98.3%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}}} \]

      2. +-commutative 98.3%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{e^{b} + e^{a}}}{e^{a}}\right)}^{-3}} \]

   6. Simplified 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}} \]

   7. Taylor expanded in b around inf 99.3%

      \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]
   8. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}} \]

      2. rem-exp-log 99.3%

         \[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}} \]

      3. div-exp 99.3%

         \[\leadsto \color{blue}{e^{a - \log \left(e^{b} + e^{a}\right)}} \]

      4. sub-neg 99.3%

         \[\leadsto e^{\color{blue}{a + \left(-\log \left(e^{b} + e^{a}\right)\right)}} \]

      5. +-commutative 99.3%

         \[\leadsto e^{\color{blue}{\left(-\log \left(e^{b} + e^{a}\right)\right) + a}} \]

      6. neg-sub0 99.3%

         \[\leadsto e^{\color{blue}{\left(0 - \log \left(e^{b} + e^{a}\right)\right)} + a} \]

      7. associate-+l- 99.3%

         \[\leadsto e^{\color{blue}{0 - \left(\log \left(e^{b} + e^{a}\right) - a\right)}} \]

      8. metadata-eval 99.3%

         \[\leadsto e^{\color{blue}{\log 1} - \left(\log \left(e^{b} + e^{a}\right) - a\right)} \]

      9. rem-log-exp 99.3%

         \[\leadsto e^{\log 1 - \left(\log \left(e^{b} + e^{a}\right) - \color{blue}{\log \left(e^{a}\right)}\right)} \]

      10. log-div 99.3%

          \[\leadsto e^{\log 1 - \color{blue}{\log \left(\frac{e^{b} + e^{a}}{e^{a}}\right)}} \]

      11. log-div 99.3%

          \[\leadsto e^{\color{blue}{\log \left(\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}\right)}} \]

      12. unpow-1 99.3%

          \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)}} \]

      13. rem-exp-log 99.3%

          \[\leadsto \color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}} \]

      14. unpow-1 99.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}} \]

      15. rem-exp-log 99.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}}{e^{a}}} \]

      16. exp-diff 99.3%

          \[\leadsto \frac{1}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right) - a}}} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{\left(-a\right) + b}}} \]

   10. Taylor expanded in b around 0 98.8%

       \[\leadsto \color{blue}{\frac{1}{1 + e^{-a}}} \]

   11. Taylor expanded in a around 0 63.6%

       \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
   12. Step-by-step derivation

       1. mul-1-neg 63.6%

          \[\leadsto \frac{1}{2 + \color{blue}{\left(-a\right)}} \]

       2. unsub-neg 63.6%

          \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   13. Simplified 63.6%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if 2e19 < (exp.f64 b)

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 26.9%

      \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]

   4. Taylor expanded in a around 0 2.8%

      \[\leadsto \color{blue}{0.5 + \left(-0.020833333333333332 \cdot {a}^{3} + 0.25 \cdot a\right)} \]

   5. Taylor expanded in a around inf 40.9%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.0001:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;e^{b} \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = exp(a);
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 98.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 98.5%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 98.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot 0.3333333333333333}} \]

      4. pow3 98.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 98.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 0.3333333333333333} \]

      6. log-div 98.5%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 98.5%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333}} \]

   5. Taylor expanded in a around inf 98.5%

      \[\leadsto e^{\color{blue}{a}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 52.4%

      \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]

   4. Taylor expanded in a around 0 52.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 52.2%

         \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

   6. Simplified 52.2%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -800:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -800.0) (exp a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -800.0) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-800.0d0)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -800.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -800.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -800.0)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -800.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -800.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -800

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 100.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot 0.3333333333333333}} \]

      4. pow3 100.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 100.0%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 0.3333333333333333} \]

      6. log-div 100.0%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 100.0%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333}} \]

   5. Taylor expanded in a around inf 100.0%

      \[\leadsto e^{\color{blue}{a}} \]

   if -800 < a

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.4%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -800:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.3% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in b around 0 64.1%

   \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]

4. Taylor expanded in a around 0 39.5%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
5. Step-by-step derivation

   1. *-commutative 39.5%

      \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

6. Simplified 39.5%

   \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

7. Final simplification 39.5%

   \[\leadsto 0.5 + a \cdot 0.25 \]
8. Add Preprocessing

Alternative 6: 40.0% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

   2. pow1/3 98.8%

      \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

   3. pow3 98.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. clear-num 98.8%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 98.8%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. pow-pow 98.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   7. metadata-eval 98.8%

      \[\leadsto {\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}}} \]

   2. +-commutative 98.3%

      \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{e^{b} + e^{a}}}{e^{a}}\right)}^{-3}} \]

6. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}} \]

7. Taylor expanded in b around inf 98.8%

   \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]
8. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{e^{b} + e^{a}}} \]

   2. rem-exp-log 98.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}} \]

   3. div-exp 98.8%

      \[\leadsto \color{blue}{e^{a - \log \left(e^{b} + e^{a}\right)}} \]

   4. sub-neg 98.8%

      \[\leadsto e^{\color{blue}{a + \left(-\log \left(e^{b} + e^{a}\right)\right)}} \]

   5. +-commutative 98.8%

      \[\leadsto e^{\color{blue}{\left(-\log \left(e^{b} + e^{a}\right)\right) + a}} \]

   6. neg-sub0 98.8%

      \[\leadsto e^{\color{blue}{\left(0 - \log \left(e^{b} + e^{a}\right)\right)} + a} \]

   7. associate-+l- 98.8%

      \[\leadsto e^{\color{blue}{0 - \left(\log \left(e^{b} + e^{a}\right) - a\right)}} \]

   8. metadata-eval 98.8%

      \[\leadsto e^{\color{blue}{\log 1} - \left(\log \left(e^{b} + e^{a}\right) - a\right)} \]

   9. rem-log-exp 98.8%

      \[\leadsto e^{\log 1 - \left(\log \left(e^{b} + e^{a}\right) - \color{blue}{\log \left(e^{a}\right)}\right)} \]

   10. log-div 98.8%

       \[\leadsto e^{\log 1 - \color{blue}{\log \left(\frac{e^{b} + e^{a}}{e^{a}}\right)}} \]

   11. log-div 98.8%

       \[\leadsto e^{\color{blue}{\log \left(\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}\right)}} \]

   12. unpow-1 98.8%

       \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)}} \]

   13. rem-exp-log 98.8%

       \[\leadsto \color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}} \]

   14. unpow-1 98.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}} \]

   15. rem-exp-log 98.8%

       \[\leadsto \frac{1}{\frac{\color{blue}{e^{\log \left(e^{b} + e^{a}\right)}}}{e^{a}}} \]

   16. exp-diff 98.8%

       \[\leadsto \frac{1}{\color{blue}{e^{\log \left(e^{b} + e^{a}\right) - a}}} \]

9. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{\left(-a\right) + b}}} \]

10. Taylor expanded in b around 0 64.9%

    \[\leadsto \color{blue}{\frac{1}{1 + e^{-a}}} \]

11. Taylor expanded in a around 0 40.0%

    \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
12. Step-by-step derivation

    1. mul-1-neg 40.0%

       \[\leadsto \frac{1}{2 + \color{blue}{\left(-a\right)}} \]

    2. unsub-neg 40.0%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

13. Simplified 40.0%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

14. Final simplification 40.0%

    \[\leadsto \frac{1}{2 - a} \]
15. Add Preprocessing

Alternative 7: 39.2% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

double code(double a, double b) {
	return 0.5;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 79.9%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]

4. Taylor expanded in b around 0 39.3%

   \[\leadsto \color{blue}{0.5} \]

5. Final simplification 39.3%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024023 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))