Quotient of sum of exps

Percentage Accurate: 99.0% → 99.7%

Time: 14.5s

Alternatives: 9

Speedup: 0.7×

• 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: 99.0% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b))))) (if (<= t_0 1.0) t_0 1.0)))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(a) / (exp(a) + exp(b))
    if (t_0 <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = Math.exp(a) / (Math.exp(a) + Math.exp(b));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.exp(a) / (math.exp(a) + math.exp(b))
	tmp = 0
	if t_0 <= 1.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = exp(a) / (exp(a) + exp(b));
	tmp = 0.0;
	if (t_0 <= 1.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], t$95$0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 1

   1. Initial program 100.0%

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

   if 1 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 0.0%

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

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. pow-to-exp 0.0%

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

      4. pow1/3 0.0%

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

      5. log-pow 0.0%

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

      6. log-div 0.0%

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

      7. add-log-exp 19.3%

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

   4. Applied egg-rr 19.3%

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

   5. Taylor expanded in a around 0 59.4%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. associate-*l* 59.4%

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

      5. log1p-def 59.4%

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

      6. metadata-eval 59.4%

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

   7. Applied egg-rr 59.4%

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

   9. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (exp (* (* 0.3333333333333333 (- a (log (+ (exp a) (exp b))))) 3.0)))

C

double code(double a, double b) {
	return exp(((0.3333333333333333 * (a - log((exp(a) + exp(b))))) * 3.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(((0.3333333333333333d0 * (a - log((exp(a) + exp(b))))) * 3.0d0))
end function

Java

public static double code(double a, double b) {
	return Math.exp(((0.3333333333333333 * (a - Math.log((Math.exp(a) + Math.exp(b))))) * 3.0));
}

Python

def code(a, b):
	return math.exp(((0.3333333333333333 * (a - math.log((math.exp(a) + math.exp(b))))) * 3.0))

Julia

function code(a, b)
	return exp(Float64(Float64(0.3333333333333333 * Float64(a - log(Float64(exp(a) + exp(b))))) * 3.0))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(((0.3333333333333333 * (a - log((exp(a) + exp(b))))) * 3.0));
end

Wolfram

code[a_, b_] := N[Exp[N[(N[(0.3333333333333333 * N[(a - N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. pow-to-exp 97.1%

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

   4. pow1/3 97.2%

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

   5. log-pow 97.6%

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

   6. log-div 97.6%

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

   7. add-log-exp 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

   \[\leadsto e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]
6. Add Preprocessing

Alternative 3: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;e^{a} \leq 1.00000000002:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (exp a)
   (if (<= (exp a) 1.00000000002) (+ 0.5 (* a 0.25)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a);
	} else if (exp(a) <= 1.00000000002) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a)
    else if (exp(a) <= 1.00000000002d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0
    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 if (Math.exp(a) <= 1.00000000002) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a)
	elif math.exp(a) <= 1.00000000002:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = exp(a);
	elseif (exp(a) <= 1.00000000002)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a);
	elseif (exp(a) <= 1.00000000002)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[Exp[a], $MachinePrecision], If[LessEqual[N[Exp[a], $MachinePrecision], 1.00000000002], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.6%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

      3. pow-to-exp 98.6%

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

      4. pow1/3 98.6%

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

      5. log-pow 98.6%

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

      6. log-div 98.6%

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

      7. add-log-exp 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in a around inf 98.7%

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

   if 0.0 < (exp.f64 a) < 1.00000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0 49.2%

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

   4. Taylor expanded in a around 0 49.2%

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

      1. *-commutative 49.2%

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

   6. Simplified 49.2%

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

   if 1.00000000002 < (exp.f64 a)

   1. Initial program 54.5%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.4%

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

      3. pow-to-exp 54.3%

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

      4. pow1/3 54.5%

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

      5. log-pow 54.5%

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

      6. log-div 54.4%

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

      7. add-log-exp 64.5%

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

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in a around 0 72.7%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 72.7%

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

      2. pow3 72.7%

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

      3. *-commutative 72.7%

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

      4. associate-*l* 72.7%

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

      5. log1p-def 72.7%

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

      6. metadata-eval 72.7%

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

   7. Applied egg-rr 72.7%

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

   9. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;e^{a} \leq 1.00000000002:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -10000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -10000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -10000.0) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-10000.0d0)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -10000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -10000.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -10000.0)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -10000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -10000.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e4

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow1/3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -1e4 < a

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -10000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.86:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.86) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.86) {
		tmp = 1.0;
	} 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 (b <= (-0.86d0)) then
        tmp = 1.0d0
    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 (b <= -0.86) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.86:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.86)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.86)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.86], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.859999999999999987

   1. Initial program 96.3%

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

      1. add-cube-cbrt 96.3%

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

      2. pow3 96.3%

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

      3. pow-to-exp 96.3%

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

      4. pow1/3 96.3%

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

      5. log-pow 96.3%

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

      6. log-div 96.3%

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

      7. add-log-exp 96.4%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. log1p-def 100.0%

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

      6. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 98.6%

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

   if -0.859999999999999987 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in b around 0 73.2%

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

   4. Taylor expanded in a around 0 39.0%

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

      1. *-commutative 39.0%

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

   6. Simplified 39.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.86:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.6% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.25) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -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 (b <= (-1.25d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.25:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.25)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.25], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.25

   1. Initial program 96.3%

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

      1. add-cube-cbrt 96.3%

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

      2. pow3 96.3%

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

      3. pow-to-exp 96.3%

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

      4. pow1/3 96.3%

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

      5. log-pow 96.3%

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

      6. log-div 96.3%

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

      7. add-log-exp 96.4%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. log1p-def 100.0%

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

      6. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 98.6%

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

   if -1.25 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0 74.5%

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

   4. Taylor expanded in b around 0 39.6%

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

      1. *-commutative 39.6%

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

   6. Simplified 39.6%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.42) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.42) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.42d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.42) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.42:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.42)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.42)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.42], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.419999999999999984

   1. Initial program 96.3%

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

      1. add-cube-cbrt 96.3%

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

      2. pow3 96.3%

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

      3. pow-to-exp 96.3%

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

      4. pow1/3 96.3%

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

      5. log-pow 96.3%

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

      6. log-div 96.3%

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

      7. add-log-exp 96.4%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. log1p-def 100.0%

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

      6. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 98.6%

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

   if -0.419999999999999984 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0 74.5%

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

   4. Taylor expanded in b around 0 40.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   5. Step-by-step derivation

      1. +-commutative 40.3%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   6. Simplified 40.3%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.75:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.75) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.75) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.75d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.75) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.75:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.75)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.75)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.75], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -0.75

   1. Initial program 96.3%

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

      1. add-cube-cbrt 96.3%

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

      2. pow3 96.3%

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

      3. pow-to-exp 96.3%

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

      4. pow1/3 96.3%

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

      5. log-pow 96.3%

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

      6. log-div 96.3%

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

      7. add-log-exp 96.4%

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{\color{blue}{\left(-0.3333333333333333 \cdot \log \left(1 + e^{b}\right)\right)} \cdot 3} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. log1p-def 100.0%

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

      6. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 98.6%

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

   if -0.75 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0 74.5%

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

   4. Taylor expanded in b around 0 38.7%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.75:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.0% 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 97.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 34.5%

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

5. Final simplification 34.5%

   \[\leadsto 0.5 \]
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 2024016 
(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))))