Quotient of sum of exps

Percentage Accurate: 99.0% → 99.0%

Time: 5.9s

Alternatives: 8

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: 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 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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 (exp(a) <= 0.0d0) 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 (Math.exp(a) <= 0.0) {
		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) <= 0.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		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 (exp(a) <= 0.0)
		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], 0.0], 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 (exp.f64 a) < 0.0

   1. Initial program 97.4%

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

      1. add-cbrt-cube 97.4%

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

      2. pow1/3 97.4%

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

      3. pow-to-exp 97.4%

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

      4. pow3 97.4%

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

      5. log-pow 97.4%

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

      6. log-div 97.4%

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

      7. add-log-exp 97.5%

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

   3. Applied egg-rr 97.5%

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

   4. Taylor expanded in a around inf 98.8%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 72.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a);
	} 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 (exp(a) <= 0.0d0) then
        tmp = exp(a)
    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 (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[Exp[a], $MachinePrecision], 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 (exp.f64 a) < 0.0

   1. Initial program 97.4%

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

      1. add-cbrt-cube 97.4%

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

      2. pow1/3 97.4%

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

      3. pow-to-exp 97.4%

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

      4. pow3 97.4%

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

      5. log-pow 97.4%

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

      6. log-div 97.4%

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

      7. add-log-exp 97.5%

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

   3. Applied egg-rr 97.5%

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

   4. Taylor expanded in a around inf 98.8%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.6%

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

   3. Taylor expanded in b around 0 64.2%

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

      1. unpow2 64.2%

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

   5. Simplified 64.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 4: 53.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -270000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -270000000.0) 0.5 (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -270000000.0) {
		tmp = 0.5;
	} 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 <= (-270000000.0d0)) then
        tmp = 0.5d0
    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 <= -270000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -270000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -270000000.0)
		tmp = 0.5;
	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 <= -270000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -270000000.0], 0.5, 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 < -2.7e8

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 18.8%

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

   if -2.7e8 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 73.3%

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

   3. Taylor expanded in b around 0 58.3%

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

      1. unpow2 58.3%

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

   5. Simplified 58.3%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -270000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 5: 52.7% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-39}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.85e-39) (+ 0.5 (* a 0.25)) (/ 1.0 (+ 2.0 (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.85e-39) {
		tmp = 0.5 + (a * 0.25);
	} 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 <= 1.85d-39) then
        tmp = 0.5d0 + (a * 0.25d0)
    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 <= 1.85e-39) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.85e-39:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (2.0 + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.85e-39)
		tmp = Float64(0.5 + Float64(a * 0.25));
	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 <= 1.85e-39)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (2.0 + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.85e-39], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 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 < 1.85000000000000007e-39

   1. Initial program 98.8%

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

   2. Taylor expanded in b around 0 80.7%

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

   3. Taylor expanded in a around 0 50.8%

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

      1. *-commutative 50.8%

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

   5. Simplified 50.8%

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

   if 1.85000000000000007e-39 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 95.2%

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

   3. Taylor expanded in b around 0 55.1%

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

      1. unpow2 55.1%

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

   5. Simplified 55.1%

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

   6. Taylor expanded in b around inf 54.9%

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

      1. unpow2 54.9%

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

   8. Simplified 54.9%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-39}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 6: 52.6% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.8e-11) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.8e-11) {
		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 <= 4.8d-11) 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 <= 4.8e-11) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.8e-11:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.8e-11)
		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 <= 4.8e-11)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.8e-11], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000002e-11

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0 81.0%

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

   3. Taylor expanded in a around 0 51.2%

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

      1. *-commutative 51.2%

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

   5. Simplified 51.2%

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

   if 4.8000000000000002e-11 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 96.2%

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

   3. Taylor expanded in b around 0 54.0%

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

      1. unpow2 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in b around inf 54.0%

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

      1. unpow2 54.0%

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

   8. Simplified 54.0%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-11}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 7: 38.7% 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 99.2%

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

2. Taylor expanded in b around 0 66.6%

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

3. Taylor expanded in a around 0 36.9%

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

   1. *-commutative 36.9%

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

5. Simplified 36.9%

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

6. Final simplification 36.9%

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

Alternative 8: 38.6% 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 99.2%

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

2. Taylor expanded in a around 0 77.5%

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

3. Taylor expanded in b around 0 36.6%

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

4. Final simplification 36.6%

   \[\leadsto 0.5 \]

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 2023229 
(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))))