Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 6.6s

Alternatives: 9

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: 98.8% 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 99.6%

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

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 99.6%

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

   3. pow3 99.6%

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

   4. clear-num 99.6%

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

   5. inv-pow 99.6%

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

   6. pow-pow 99.6%

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

   7. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1/3 99.0%

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

   2. +-commutative 99.0%

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

6. Simplified 99.0%

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

   1. expm1-log1p-u 99.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

   2. expm1-udef 99.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

   3. pow1/3 99.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

   4. pow-pow 99.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

   5. metadata-eval 99.5%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

   2. expm1-log1p 99.6%

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

   3. unpow-1 99.6%

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

   4. rem-exp-log 99.6%

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

   5. exp-diff 100.0%

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

   6. sub-neg 100.0%

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

   7. prod-exp 99.6%

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

   8. rem-exp-log 99.6%

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

   9. *-commutative 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-lft-in 77.3%

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

   12. rec-exp 77.3%

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

   13. lft-mult-inverse 99.6%

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

   14. prod-exp 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e-15) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-15) {
		tmp = 1.0 / (1.0 + 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 <= (-5.8d-15)) then
        tmp = 1.0d0 / (1.0d0 + 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 <= -5.8e-15) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.8e-15:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.8e-15)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-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 <= -5.8e-15)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.8e-15], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.80000000000000037e-15

   1. Initial program 100.0%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.9%

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

      3. pow3 100.0%

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

      4. clear-num 99.9%

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

      5. inv-pow 99.9%

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

      6. pow-pow 99.9%

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

      7. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow1/3 99.9%

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

      2. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 99.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 99.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 99.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 99.9%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 99.9%

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

      3. unpow-1 99.9%

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

      4. rem-exp-log 99.9%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 100.0%

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

      8. rem-exp-log 100.0%

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

      9. *-commutative 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 5.0%

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

      12. rec-exp 5.0%

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

      13. lft-mult-inverse 100.0%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 100.0%

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

   if -5.80000000000000037e-15 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0) 0.0 (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		tmp = 0.0;
	} 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 <= (-720.0d0)) then
        tmp = 0.0d0
    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 <= -720.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -720.0)
		tmp = 0.0;
	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 <= -720.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720.0], 0.0, N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -720

   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. pow3 100.0%

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

      4. clear-num 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/3 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. unpow-1 100.0%

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

      4. rem-exp-log 100.0%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 100.0%

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

      8. rem-exp-log 100.0%

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

      9. *-commutative 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 0.0%

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

      12. rec-exp 0.0%

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

      13. lft-mult-inverse 100.0%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 100.0%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]

   if -720 < a

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e-15) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-15) {
		tmp = exp(a) / (a + 2.0);
	} 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 <= (-5.8d-15)) then
        tmp = exp(a) / (a + 2.0d0)
    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 <= -5.8e-15) {
		tmp = Math.exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.8e-15:
		tmp = math.exp(a) / (a + 2.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.8e-15)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	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 <= -5.8e-15)
		tmp = exp(a) / (a + 2.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.8e-15], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.80000000000000037e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 99.5%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   5. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   6. Simplified 99.5%

      \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   if -5.80000000000000037e-15 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.05) 1.0 (if (<= b 1.4) (+ 0.5 (* a 0.25)) 0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.05d0)) then
        tmp = 1.0d0
    else if (b <= 1.4d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.05:
		tmp = 1.0
	elif b <= 1.4:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.05)
		tmp = 1.0;
	elseif (b <= 1.4)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.05)
		tmp = 1.0;
	elseif (b <= 1.4)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.05], 1.0, If[LessEqual[b, 1.4], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000004

   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. pow3 100.0%

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

      4. clear-num 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/3 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. unpow-1 100.0%

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

      4. rem-exp-log 100.0%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 100.0%

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

      8. rem-exp-log 100.0%

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

      9. *-commutative 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. rec-exp 100.0%

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

      13. lft-mult-inverse 100.0%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 18.8%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.05000000000000004 < b < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 98.9%

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

   4. Taylor expanded in a around 0 69.7%

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

      1. *-commutative 69.7%

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

   6. Simplified 69.7%

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

   if 1.3999999999999999 < b

   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. pow3 98.5%

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

      4. clear-num 98.5%

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

      5. inv-pow 98.5%

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

      6. pow-pow 98.5%

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

      7. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. unpow1/3 98.5%

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

      2. +-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 98.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 98.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 98.5%

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

      3. unpow-1 98.5%

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

      4. rem-exp-log 98.5%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 98.5%

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

      8. rem-exp-log 98.5%

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

      9. *-commutative 98.5%

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

      10. +-commutative 98.5%

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

      11. distribute-lft-in 75.4%

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

      12. rec-exp 75.4%

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

      13. lft-mult-inverse 98.5%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 25.5%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.1) 1.0 (if (<= b 1.75) 0.5 0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else if (b <= 1.75) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else if (b <= 1.75d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else if (b <= 1.75) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	elif b <= 1.75:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	elseif (b <= 1.75)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	elseif (b <= 1.75)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], 1.0, If[LessEqual[b, 1.75], 0.5, 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1000000000000001

   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. pow3 100.0%

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

      4. clear-num 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/3 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. unpow-1 100.0%

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

      4. rem-exp-log 100.0%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 100.0%

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

      8. rem-exp-log 100.0%

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

      9. *-commutative 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. rec-exp 100.0%

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

      13. lft-mult-inverse 100.0%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 18.8%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.1000000000000001 < b < 1.75

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 69.8%

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

   4. Taylor expanded in b around 0 68.7%

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

   if 1.75 < b

   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. pow3 98.5%

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

      4. clear-num 98.5%

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

      5. inv-pow 98.5%

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

      6. pow-pow 98.5%

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

      7. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. unpow1/3 98.5%

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

      2. +-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 98.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 98.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 98.5%

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

      3. unpow-1 98.5%

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

      4. rem-exp-log 98.5%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 98.5%

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

      8. rem-exp-log 98.5%

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

      9. *-commutative 98.5%

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

      10. +-commutative 98.5%

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

      11. distribute-lft-in 75.4%

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

      12. rec-exp 75.4%

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

      13. lft-mult-inverse 98.5%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 25.5%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.7% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -3.4e-15) 0.16666666666666666 0.0))

C

double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-15) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3.4d-15)) then
        tmp = 0.16666666666666666d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-15) {
		tmp = 0.16666666666666666;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -3.4e-15:
		tmp = 0.16666666666666666
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -3.4e-15)
		tmp = 0.16666666666666666;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -3.4e-15)
		tmp = 0.16666666666666666;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3.4e-15], 0.16666666666666666, 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -3.4e-15

   1. Initial program 100.0%

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

      1. add-cbrt-cube 99.9%

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

      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. pow3 100.0%

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

      4. clear-num 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-pow 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1/3 99.9%

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

      2. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. unpow-1 100.0%

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

      4. rem-exp-log 100.0%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 100.0%

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

      8. rem-exp-log 100.0%

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

      9. *-commutative 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. rec-exp 100.0%

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

      13. lft-mult-inverse 100.0%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 22.3%

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

   13. Applied egg-rr 16.6%

       \[\leadsto \color{blue}{0.16666666666666666} \]

   if -3.4e-15 < b

   1. Initial program 99.5%

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

      1. add-cbrt-cube 98.8%

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

      2. pow1/3 99.5%

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

      3. pow3 99.5%

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

      4. clear-num 99.5%

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

      5. inv-pow 99.5%

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

      6. pow-pow 99.5%

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

      7. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/3 98.8%

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

      2. +-commutative 98.8%

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

   6. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 99.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 99.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 99.4%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 99.5%

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

      3. unpow-1 99.5%

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

      4. rem-exp-log 99.5%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 99.5%

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

      8. rem-exp-log 99.5%

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

      9. *-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. distribute-lft-in 72.0%

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

      12. rec-exp 72.0%

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

      13. lft-mult-inverse 99.5%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 76.2%

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

   13. Applied egg-rr 53.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 2.05) 0.5 0.0))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.05d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.05:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.05)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.05)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.05], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 2.0499999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 77.1%

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

   4. Taylor expanded in b around 0 56.7%

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

   if 2.0499999999999998 < b

   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. pow3 98.5%

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

      4. clear-num 98.5%

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

      5. inv-pow 98.5%

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

      6. pow-pow 98.5%

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

      7. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. unpow1/3 98.5%

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

      2. +-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

      2. expm1-udef 98.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

      3. pow1/3 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

      4. pow-pow 98.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

      5. metadata-eval 98.5%

         \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 98.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-log1p 98.5%

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

      3. unpow-1 98.5%

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

      4. rem-exp-log 98.5%

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

      5. exp-diff 100.0%

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

      6. sub-neg 100.0%

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

      7. prod-exp 98.5%

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

      8. rem-exp-log 98.5%

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

      9. *-commutative 98.5%

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

      10. +-commutative 98.5%

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

      11. distribute-lft-in 75.4%

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

      12. rec-exp 75.4%

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

      13. lft-mult-inverse 98.5%

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

      14. prod-exp 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around 0 25.5%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.6% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.0)

C

double code(double a, double b) {
	return 0.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

public static double code(double a, double b) {
	return 0.0;
}

Python

def code(a, b):
	return 0.0

Julia

function code(a, b)
	return 0.0
end

MATLAB

function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

code[a_, b_] := 0.0

Derivation

1. Initial program 99.6%

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

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 99.6%

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

   3. pow3 99.6%

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

   4. clear-num 99.6%

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

   5. inv-pow 99.6%

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

   6. pow-pow 99.6%

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

   7. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1/3 99.0%

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

   2. +-commutative 99.0%

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

6. Simplified 99.0%

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

   1. expm1-log1p-u 99.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)\right)} \]

   2. expm1-udef 99.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}\right)} - 1} \]

   3. pow1/3 99.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}}\right)} - 1 \]

   4. pow-pow 99.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

   5. metadata-eval 99.5%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{\color{blue}{-1}}\right)} - 1 \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-1}\right)\right)} \]

   2. expm1-log1p 99.6%

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

   3. unpow-1 99.6%

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

   4. rem-exp-log 99.6%

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

   5. exp-diff 100.0%

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

   6. sub-neg 100.0%

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

   7. prod-exp 99.6%

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

   8. rem-exp-log 99.6%

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

   9. *-commutative 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-lft-in 77.3%

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

   12. rec-exp 77.3%

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

   13. lft-mult-inverse 99.6%

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

   14. prod-exp 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in b around 0 65.8%

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

13. Applied egg-rr 43.6%

    \[\leadsto \color{blue}{0} \]

14. Final simplification 43.6%

    \[\leadsto 0 \]
15. 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 2024011 
(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))))