Quotient of sum of exps

Percentage Accurate: 99.1% → 99.1%

Time: 4.5s

Alternatives: 10

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: 99.1% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;e^{a} \cdot \frac{1}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999998)
   (* (exp a) (/ 1.0 (+ 1.0 (exp a))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999998) {
		tmp = exp(a) * (1.0 / (1.0 + exp(a)));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.99999998d0) then
        tmp = exp(a) * (1.0d0 / (1.0d0 + exp(a)))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999998) {
		tmp = Math.exp(a) * (1.0 / (1.0 + Math.exp(a)));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999998:
		tmp = math.exp(a) * (1.0 / (1.0 + math.exp(a)))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999998)
		tmp = Float64(exp(a) * Float64(1.0 / Float64(1.0 + exp(a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99999998)
		tmp = exp(a) * (1.0 / (1.0 + exp(a)));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999998], N[(N[Exp[a], $MachinePrecision] * N[(1.0 / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999980000000011

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if 0.999999980000000011 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;e^{a} \cdot \frac{1}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999998)
   (/ (exp a) (+ 1.0 (exp a)))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999998) {
		tmp = exp(a) / (1.0 + exp(a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.99999998d0) then
        tmp = exp(a) / (1.0d0 + exp(a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999998) {
		tmp = Math.exp(a) / (1.0 + Math.exp(a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999998:
		tmp = math.exp(a) / (1.0 + math.exp(a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999998)
		tmp = Float64(exp(a) / Float64(1.0 + exp(a)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99999998)
		tmp = exp(a) / (1.0 + exp(a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999998], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999980000000011

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   if 0.999999980000000011 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 99.1% 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 5: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1) (exp a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.1d0) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = exp(a);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.1], N[Exp[a], $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 99.0%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 6: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.1d0) then
        tmp = exp(a)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = exp(a);
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = exp(a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.1], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 99.0%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 58.0%

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

   3. Taylor expanded in a around 0 57.8%

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

      1. *-commutative 57.8%

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

   5. Simplified 57.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 7: 80.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.14 \cdot 10^{+92} \lor \neg \left(b \leq 6.8 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.06)
   (exp a)
   (if (or (<= b 1.14e+92) (not (<= b 6.8e+190)))
     (/ (exp a) 2.0)
     (* -0.020833333333333332 (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = exp(a);
	} else if ((b <= 1.14e+92) || !(b <= 6.8e+190)) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = -0.020833333333333332 * pow(a, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.06d0)) then
        tmp = exp(a)
    else if ((b <= 1.14d+92) .or. (.not. (b <= 6.8d+190))) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (-0.020833333333333332d0) * (a ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = Math.exp(a);
	} else if ((b <= 1.14e+92) || !(b <= 6.8e+190)) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.06:
		tmp = math.exp(a)
	elif (b <= 1.14e+92) or not (b <= 6.8e+190):
		tmp = math.exp(a) / 2.0
	else:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.06)
		tmp = exp(a);
	elseif ((b <= 1.14e+92) || !(b <= 6.8e+190))
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.06)
		tmp = exp(a);
	elseif ((b <= 1.14e+92) || ~((b <= 6.8e+190)))
		tmp = exp(a) / 2.0;
	else
		tmp = -0.020833333333333332 * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.06], N[Exp[a], $MachinePrecision], If[Or[LessEqual[b, 1.14e+92], N[Not[LessEqual[b, 6.8e+190]], $MachinePrecision]], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0600000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 98.4%

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

   if -1.0600000000000001 < b < 1.13999999999999993e92 or 6.7999999999999999e190 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 85.8%

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

   3. Taylor expanded in a around 0 84.4%

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

   if 1.13999999999999993e92 < b < 6.7999999999999999e190

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 25.1%

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

   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 43.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.14 \cdot 10^{+92} \lor \neg \left(b \leq 6.8 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \]

Alternative 8: 80.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) (exp a) (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = exp(a);
	} else {
		tmp = exp(a) / 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 <= (-1.1d0)) then
        tmp = exp(a)
    else
        tmp = exp(a) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = Math.exp(a);
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = math.exp(a)
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = exp(a);
	else
		tmp = Float64(exp(a) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = exp(a);
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], N[Exp[a], $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 98.4%

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

   if -1.1000000000000001 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 79.7%

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

   3. Taylor expanded in a around 0 78.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 9: 39.5% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in b around 0 70.1%

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

3. Taylor expanded in a around 0 41.7%

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

   1. *-commutative 41.7%

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

5. Simplified 41.7%

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

6. Final simplification 41.7%

   \[\leadsto 0.5 + a \cdot 0.25 \]

Alternative 10: 39.5% 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 99.6%

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

2. Taylor expanded in a around 0 81.5%

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

3. Taylor expanded in b around 0 41.4%

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

4. Final simplification 41.4%

   \[\leadsto 0.5 \]

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 2023240 
(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))))