Quotient of sum of exps

Percentage Accurate: 99.0% → 98.5%

Time: 8.8s

Alternatives: 14

Speedup: 1.0×

• 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: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999986:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999986)
   (/ (exp a) (+ (exp a) 1.0))
   (exp (- (log1p (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999986) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = exp(-log1p(exp(b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.9999986) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.exp(-Math.log1p(Math.exp(b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9999986:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = math.exp(-math.log1p(math.exp(b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999986)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = exp(Float64(-log1p(exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999986], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999859999999996

   1. Initial program 98.3%

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

   2. Taylor expanded in b around 0 100.0%

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

   if 0.99999859999999996 < (exp.f64 a)

   1. Initial program 98.4%

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

      1. add-exp-log 98.4%

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

      2. div-exp 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. log1p-def 98.7%

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

   6. Simplified 98.7%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999986:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. add-exp-log 98.4%

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

   2. div-exp 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999986:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999986)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999986) {
		tmp = exp(a) / (exp(a) + 1.0);
	} 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 (exp(a) <= 0.9999986d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    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 (Math.exp(a) <= 0.9999986) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9999986:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999986)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	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 (exp(a) <= 0.9999986)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999986], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999859999999996

   1. Initial program 98.3%

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

   2. Taylor expanded in b around 0 100.0%

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

   if 0.99999859999999996 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999986:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4: 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]

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 5: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-76) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-76) {
		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 (exp(a) <= 2d-76) 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 (Math.exp(a) <= 2e-76) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-76:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-76)
		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 (exp(a) <= 2e-76)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-76], 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 (exp.f64 a) < 1.99999999999999985e-76

   1. Initial program 98.2%

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

      1. add-exp-log 98.2%

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

      2. div-exp 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in a around inf 100.0%

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

   if 1.99999999999999985e-76 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6: 71.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-110}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-157}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 380000:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.05e-7)
   1.0
   (if (<= b -5.6e-110)
     (+ 0.5 (* a 0.25))
     (if (<= b -3.9e-157)
       (exp a)
       (if (<= b 380000.0)
         (+ 0.5 (* b -0.25))
         (if (<= b 1.35e+154)
           (* -0.020833333333333332 (pow a 3.0))
           (/ 2.0 (* b b))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= -5.6e-110) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -3.9e-157) {
		tmp = exp(a);
	} else if (b <= 380000.0) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * pow(a, 3.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.05d-7)) then
        tmp = 1.0d0
    else if (b <= (-5.6d-110)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-3.9d-157)) then
        tmp = exp(a)
    else if (b <= 380000.0d0) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else if (b <= 1.35d+154) then
        tmp = (-0.020833333333333332d0) * (a ** 3.0d0)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= -5.6e-110) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -3.9e-157) {
		tmp = Math.exp(a);
	} else if (b <= 380000.0) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.05e-7:
		tmp = 1.0
	elif b <= -5.6e-110:
		tmp = 0.5 + (a * 0.25)
	elif b <= -3.9e-157:
		tmp = math.exp(a)
	elif b <= 380000.0:
		tmp = 0.5 + (b * -0.25)
	elif b <= 1.35e+154:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= -5.6e-110)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -3.9e-157)
		tmp = exp(a);
	elseif (b <= 380000.0)
		tmp = Float64(0.5 + Float64(b * -0.25));
	elseif (b <= 1.35e+154)
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= -5.6e-110)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -3.9e-157)
		tmp = exp(a);
	elseif (b <= 380000.0)
		tmp = 0.5 + (b * -0.25);
	elseif (b <= 1.35e+154)
		tmp = -0.020833333333333332 * (a ^ 3.0);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.05e-7], 1.0, If[LessEqual[b, -5.6e-110], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.9e-157], N[Exp[a], $MachinePrecision], If[LessEqual[b, 380000.0], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if b < -4.04999999999999987e-7

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in a around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. log1p-def 96.7%

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

   6. Simplified 96.7%

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

      1. exp-neg 96.6%

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

      2. log1p-udef 96.6%

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

      3. add-exp-log 96.6%

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

      4. add-sqr-sqrt 96.6%

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

      5. associate-/r* 96.6%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 95.4%

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

   10. Simplified 95.4%

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

   if -4.04999999999999987e-7 < b < -5.6000000000000001e-110

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0 99.9%

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

   3. Taylor expanded in a around 0 90.5%

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

   if -5.6000000000000001e-110 < b < -3.89999999999999999e-157

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 81.9%

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

   if -3.89999999999999999e-157 < b < 3.8e5

   1. Initial program 97.9%

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

   2. Taylor expanded in a around 0 70.1%

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

   3. Taylor expanded in b around 0 68.2%

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

      1. *-commutative 68.2%

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

   5. Simplified 68.2%

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

   if 3.8e5 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 21.9%

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

   3. Taylor expanded in a around 0 2.8%

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

   4. Taylor expanded in a around inf 45.4%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{1}{2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in b around inf 100.0%

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-110}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-157}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 380000:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 7: 70.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-110}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-157}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 0.002:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+142}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.05e-7)
   1.0
   (if (<= b -5.6e-110)
     (+ 0.5 (* a 0.25))
     (if (<= b -3.9e-157)
       (exp a)
       (if (<= b 0.002)
         (+ 0.5 (* b -0.25))
         (if (<= b 1.08e+142) (exp a) (/ 2.0 (* b b))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= -5.6e-110) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -3.9e-157) {
		tmp = exp(a);
	} else if (b <= 0.002) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 1.08e+142) {
		tmp = exp(a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.05d-7)) then
        tmp = 1.0d0
    else if (b <= (-5.6d-110)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-3.9d-157)) then
        tmp = exp(a)
    else if (b <= 0.002d0) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else if (b <= 1.08d+142) then
        tmp = exp(a)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= -5.6e-110) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -3.9e-157) {
		tmp = Math.exp(a);
	} else if (b <= 0.002) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 1.08e+142) {
		tmp = Math.exp(a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.05e-7:
		tmp = 1.0
	elif b <= -5.6e-110:
		tmp = 0.5 + (a * 0.25)
	elif b <= -3.9e-157:
		tmp = math.exp(a)
	elif b <= 0.002:
		tmp = 0.5 + (b * -0.25)
	elif b <= 1.08e+142:
		tmp = math.exp(a)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= -5.6e-110)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -3.9e-157)
		tmp = exp(a);
	elseif (b <= 0.002)
		tmp = Float64(0.5 + Float64(b * -0.25));
	elseif (b <= 1.08e+142)
		tmp = exp(a);
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= -5.6e-110)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -3.9e-157)
		tmp = exp(a);
	elseif (b <= 0.002)
		tmp = 0.5 + (b * -0.25);
	elseif (b <= 1.08e+142)
		tmp = exp(a);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.05e-7], 1.0, If[LessEqual[b, -5.6e-110], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.9e-157], N[Exp[a], $MachinePrecision], If[LessEqual[b, 0.002], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e+142], N[Exp[a], $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4.04999999999999987e-7

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in a around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. log1p-def 96.7%

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

   6. Simplified 96.7%

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

      1. exp-neg 96.6%

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

      2. log1p-udef 96.6%

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

      3. add-exp-log 96.6%

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

      4. add-sqr-sqrt 96.6%

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

      5. associate-/r* 96.6%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 95.4%

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

   10. Simplified 95.4%

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

   if -4.04999999999999987e-7 < b < -5.6000000000000001e-110

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0 99.9%

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

   3. Taylor expanded in a around 0 90.5%

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

   if -5.6000000000000001e-110 < b < -3.89999999999999999e-157 or 2e-3 < b < 1.08e142

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 38.6%

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

   if -3.89999999999999999e-157 < b < 2e-3

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0 70.2%

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

   3. Taylor expanded in b around 0 70.2%

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

      1. *-commutative 70.2%

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

   5. Simplified 70.2%

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

   if 1.08e142 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 96.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. unpow2 96.9%

         \[\leadsto \frac{1}{2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   5. Simplified 96.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in b around inf 96.9%

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 96.9%

         \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   8. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-110}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-157}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 0.002:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+142}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 8: 67.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.4) 1.0 (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.4d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (b + (0.5d0 * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.4:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.4)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.4)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.4], 1.0, N[(1.0 / N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in a around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. log1p-def 98.3%

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

   6. Simplified 98.3%

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

      1. exp-neg 98.3%

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

      2. log1p-udef 98.3%

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

      3. add-exp-log 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-/r* 98.3%

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

   8. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 97.0%

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

   10. Simplified 97.0%

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

   if -1.3999999999999999 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 79.6%

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

   3. Taylor expanded in b around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto \frac{1}{2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   5. Simplified 63.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 9: 67.0% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.76:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.76) 1.0 (/ 1.0 (+ 2.0 (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (0.5 * (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.76d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.76:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.76)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.76)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.76], 1.0, N[(1.0 / N[(2.0 + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.76000000000000001

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in a around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. log1p-def 98.3%

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

   6. Simplified 98.3%

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

      1. exp-neg 98.3%

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

      2. log1p-udef 98.3%

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

      3. add-exp-log 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-/r* 98.3%

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

   8. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 97.0%

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

   10. Simplified 97.0%

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

   if -0.76000000000000001 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 79.6%

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

   3. Taylor expanded in b around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. unpow2 63.9%

         \[\leadsto \frac{1}{2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   5. Simplified 63.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in b around inf 63.4%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{1}{2 + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}} \]

   8. Simplified 63.4%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.76:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 10: 67.3% accurate, 33.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.82:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.05e-7) 1.0 (if (<= b 1.82) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= 1.82) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.05d-7)) then
        tmp = 1.0d0
    else if (b <= 1.82d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		tmp = 1.0;
	} else if (b <= 1.82) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.05e-7:
		tmp = 1.0
	elif b <= 1.82:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= 1.82)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.05e-7)
		tmp = 1.0;
	elseif (b <= 1.82)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.05e-7], 1.0, If[LessEqual[b, 1.82], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.04999999999999987e-7

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in a around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. log1p-def 96.7%

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

   6. Simplified 96.7%

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

      1. exp-neg 96.6%

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

      2. log1p-udef 96.6%

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

      3. add-exp-log 96.6%

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

      4. add-sqr-sqrt 96.6%

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

      5. associate-/r* 96.6%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 95.4%

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

   10. Simplified 95.4%

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

   if -4.04999999999999987e-7 < b < 1.82000000000000006

   1. Initial program 98.5%

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

   2. Taylor expanded in b around 0 97.7%

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

   3. Taylor expanded in a around 0 71.7%

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

   if 1.82000000000000006 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.5%

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

   3. Taylor expanded in b around 0 47.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. unpow2 47.9%

         \[\leadsto \frac{1}{2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]

   5. Simplified 47.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in b around inf 47.9%

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 47.9%

         \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   8. Simplified 47.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.82:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 11: 54.0% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -4.05e-7) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.05e-7) {
		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 <= (-4.05d-7)) 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 <= -4.05e-7) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.05e-7:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.05e-7)
		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 <= -4.05e-7)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.05e-7], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.04999999999999987e-7

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.6%

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

   3. Applied egg-rr 96.6%

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

   4. Taylor expanded in a around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. log1p-def 96.7%

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

   6. Simplified 96.7%

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

      1. exp-neg 96.6%

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

      2. log1p-udef 96.6%

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

      3. add-exp-log 96.6%

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

      4. add-sqr-sqrt 96.6%

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

      5. associate-/r* 96.6%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 95.4%

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

   10. Simplified 95.4%

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

   if -4.04999999999999987e-7 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0 77.4%

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

   3. Taylor expanded in a around 0 50.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 12: 54.8% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.68) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.68) {
		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 <= (-0.68d0)) 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 <= -0.68) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.68:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.68)
		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 <= -0.68)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.68], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.680000000000000049

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in a around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. log1p-def 98.3%

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

   6. Simplified 98.3%

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

      1. exp-neg 98.3%

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

      2. log1p-udef 98.3%

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

      3. add-exp-log 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-/r* 98.3%

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

   8. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 97.0%

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

   10. Simplified 97.0%

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

   if -0.680000000000000049 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 79.6%

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

   3. Taylor expanded in b around 0 50.6%

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

      1. +-commutative 50.6%

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

   5. Simplified 50.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \]

Alternative 13: 53.8% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.68:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.68) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.68) {
		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 <= (-0.68d0)) 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 <= -0.68) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.68:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.68)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.68)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.68], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -0.680000000000000049

   1. Initial program 96.5%

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

      1. add-exp-log 96.5%

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

      2. div-exp 96.5%

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

   3. Applied egg-rr 96.5%

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

   4. Taylor expanded in a around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. log1p-def 98.3%

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

   6. Simplified 98.3%

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

      1. exp-neg 98.3%

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

      2. log1p-udef 98.3%

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

      3. add-exp-log 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-/r* 98.3%

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

   8. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + e^{b}}}{\sqrt{1 + e^{b}}}} \]
   9. Step-by-step derivation

      1. *-inverses 97.0%

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

   10. Simplified 97.0%

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

   if -0.680000000000000049 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 79.6%

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

   3. Taylor expanded in b around 0 49.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.68:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14: 39.0% 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 98.4%

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

2. Taylor expanded in a around 0 83.8%

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

3. Taylor expanded in b around 0 42.6%

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

4. Final simplification 42.6%

   \[\leadsto 0.5 \]

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 2023182 
(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))))