Quotient of sum of exps

Percentage Accurate: 99.1% → 99.2%

Time: 7.1s

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: 99.1% 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: 99.2% 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 99.6%

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

   1. add-exp-log 99.6%

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

   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. Final simplification 100.0%

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

Alternative 2: 65.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1.0)
		tmp = Float64(exp(a) / Float64(exp(a) + Float64(b + 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) <= 1.0)
		tmp = exp(a) / (exp(a) + (b + 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], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $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) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

   4. Simplified 67.9%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   2. Taylor expanded in a around 0 86.0%

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

4. Final simplification 68.6%

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

Alternative 3: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		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) <= 1.0:
		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) <= 1.0)
		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) <= 1.0)
		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], 1.0], 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) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 69.3%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   2. Taylor expanded in a around 0 86.0%

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

4. Final simplification 69.9%

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

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: 64.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] * N[(1.0 / N[(a + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $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) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in a around 0 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(a + b\right)}} \]
   6. Step-by-step derivation

      1. associate-+r+ 67.8%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]

   7. Simplified 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]
   8. Step-by-step derivation

      1. clear-num 67.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(2 + a\right) + b}{e^{a}}}} \]

      2. associate-/r/ 67.8%

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

      3. +-commutative 67.8%

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

      4. associate-+l+ 67.8%

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

   9. Applied egg-rr 67.8%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   2. Taylor expanded in a around 0 86.0%

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

4. Final simplification 68.4%

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

Alternative 6: 64.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(b + N[(a + 2.0), $MachinePrecision]), $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) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in a around 0 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(a + b\right)}} \]
   6. Step-by-step derivation

      1. associate-+r+ 67.8%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]

   7. Simplified 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   2. Taylor expanded in a around 0 86.0%

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

4. Final simplification 68.4%

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

Alternative 7: 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 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 b around 0 100.0%

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

      1. log1p-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

   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 99.0%

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

Alternative 8: 66.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $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) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

   4. Simplified 67.9%

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

   5. Taylor expanded in a around 0 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(a + b\right)}} \]
   6. Step-by-step derivation

      1. associate-+r+ 67.8%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]

   7. Simplified 67.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{\left(2 + a\right) + b}} \]

   8. Taylor expanded in b around 0 69.2%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   2. Taylor expanded in a around 0 86.0%

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

4. Final simplification 69.8%

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

Alternative 9: 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 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 b around 0 100.0%

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

      1. log1p-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 98.6%

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

   3. Taylor expanded in b around 0 68.5%

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

      1. unpow2 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 77.0%

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

Alternative 10: 60.4% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 390:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+100}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.020833333333333332\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0 - b}{b \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (* b b))))
   (if (<= b 390.0)
     (+ 0.5 (* a 0.25))
     (if (<= b 4.7e+100)
       (* (* a (* a a)) -0.020833333333333332)
       (if (<= b 1.35e+154) (/ (- t_0 b) (* b t_0)) (/ -2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= 390.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 4.7e+100) {
		tmp = (a * (a * a)) * -0.020833333333333332;
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (b * b)
    if (b <= 390.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 4.7d+100) then
        tmp = (a * (a * a)) * (-0.020833333333333332d0)
    else if (b <= 1.35d+154) then
        tmp = (t_0 - b) / (b * t_0)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= 390.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 4.7e+100) {
		tmp = (a * (a * a)) * -0.020833333333333332;
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * (b * b)
	tmp = 0
	if b <= 390.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 4.7e+100:
		tmp = (a * (a * a)) * -0.020833333333333332
	elif b <= 1.35e+154:
		tmp = (t_0 - b) / (b * t_0)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(b * b))
	tmp = 0.0
	if (b <= 390.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 4.7e+100)
		tmp = Float64(Float64(a * Float64(a * a)) * -0.020833333333333332);
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(t_0 - b) / Float64(b * t_0));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * (b * b);
	tmp = 0.0;
	if (b <= 390.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 4.7e+100)
		tmp = (a * (a * a)) * -0.020833333333333332;
	elseif (b <= 1.35e+154)
		tmp = (t_0 - b) / (b * t_0);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 390.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.7e+100], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(t$95$0 - b), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 390

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 81.4%

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

   3. Taylor expanded in a around 0 57.5%

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

      1. *-commutative 57.5%

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

   5. Simplified 57.5%

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

   if 390 < b < 4.7e100

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 51.6%

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

   3. Taylor expanded in a around 0 2.4%

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

   4. Taylor expanded in a around inf 26.9%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]
   5. Step-by-step derivation

      1. *-commutative 26.9%

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

   6. Simplified 26.9%

      \[\leadsto \color{blue}{{a}^{3} \cdot -0.020833333333333332} \]
   7. Step-by-step derivation

      1. unpow3 26.9%

         \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot -0.020833333333333332 \]

   8. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot -0.020833333333333332 \]

   if 4.7e100 < b < 1.35000000000000003e154

   1. Initial program 92.3%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 4.3%

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

      1. +-commutative 4.3%

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

   5. Simplified 4.3%

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

   6. Taylor expanded in b around inf 4.3%

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

      1. associate-*r/ 4.3%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 4.3%

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

      3. unpow2 4.3%

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

   8. Simplified 4.3%

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

      1. clear-num 4.3%

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

      2. frac-sub 92.6%

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

      3. *-un-lft-identity 92.6%

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

      4. div-inv 92.6%

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

      5. metadata-eval 92.6%

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

      6. *-commutative 92.6%

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

      7. *-un-lft-identity 92.6%

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

      8. div-inv 92.6%

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

      9. metadata-eval 92.6%

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

   10. Applied egg-rr 92.6%

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

   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 8.1%

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

      1. +-commutative 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in b around inf 8.1%

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

      1. associate-*r/ 8.1%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 8.1%

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

      3. unpow2 8.1%

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

   8. Simplified 8.1%

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

   9. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 390:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+100}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.020833333333333332\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(b \cdot b\right) - b}{b \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 11: 58.0% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 390:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+152}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 390.0)
   (+ 0.5 (* a 0.25))
   (if (<= b 1.8e+152)
     (* (* a (* a a)) -0.020833333333333332)
     (/ -2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 390.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.8e+152) {
		tmp = (a * (a * a)) * -0.020833333333333332;
	} 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 <= 390.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.8d+152) then
        tmp = (a * (a * a)) * (-0.020833333333333332d0)
    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 <= 390.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.8e+152) {
		tmp = (a * (a * a)) * -0.020833333333333332;
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 390.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.8e+152:
		tmp = (a * (a * a)) * -0.020833333333333332
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 390.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.8e+152)
		tmp = Float64(Float64(a * Float64(a * a)) * -0.020833333333333332);
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 390.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.8e+152)
		tmp = (a * (a * a)) * -0.020833333333333332;
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 390.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+152], N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 390

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 81.4%

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

   3. Taylor expanded in a around 0 57.5%

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

      1. *-commutative 57.5%

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

   5. Simplified 57.5%

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

   if 390 < b < 1.7999999999999999e152

   1. Initial program 96.8%

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

   2. Taylor expanded in b around 0 43.7%

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

   3. Taylor expanded in a around 0 2.5%

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

   4. Taylor expanded in a around inf 27.9%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]
   5. Step-by-step derivation

      1. *-commutative 27.9%

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

   6. Simplified 27.9%

      \[\leadsto \color{blue}{{a}^{3} \cdot -0.020833333333333332} \]
   7. Step-by-step derivation

      1. unpow3 27.9%

         \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot -0.020833333333333332 \]

   8. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \cdot -0.020833333333333332 \]

   if 1.7999999999999999e152 < 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 7.9%

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

      1. +-commutative 7.9%

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

   5. Simplified 7.9%

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

   6. Taylor expanded in b around inf 7.9%

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

      1. associate-*r/ 7.9%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 7.9%

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

      3. unpow2 7.9%

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

   8. Simplified 7.9%

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

   9. Taylor expanded in b around 0 96.2%

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

       1. unpow2 96.2%

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

   11. Simplified 96.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 390:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+152}:\\ \;\;\;\;\left(a \cdot \left(a \cdot a\right)\right) \cdot -0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 12: 52.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 600.0) (+ 0.5 (* a 0.25)) (/ -2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 600.0) {
		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 <= 600.0d0) 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 <= 600.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 600.0:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 600.0)
		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 <= 600.0)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 600.0], 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 < 600

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 81.4%

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

   3. Taylor expanded in a around 0 57.5%

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

      1. *-commutative 57.5%

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

   5. Simplified 57.5%

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

   if 600 < b

   1. Initial program 98.7%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 6.3%

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

      1. +-commutative 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in b around inf 6.3%

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

      1. associate-*r/ 6.3%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 6.3%

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

      3. unpow2 6.3%

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

   8. Simplified 6.3%

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

   9. Taylor expanded in b around 0 59.0%

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

       1. unpow2 59.0%

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

   11. Simplified 59.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 600:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 13: 38.9% 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 68.0%

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

3. Taylor expanded in a around 0 41.3%

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

   1. *-commutative 41.3%

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

5. Simplified 41.3%

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

6. Final simplification 41.3%

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

Alternative 14: 38.8% 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 82.7%

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

3. Taylor expanded in b around 0 40.9%

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

4. Final simplification 40.9%

   \[\leadsto 0.5 \]

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 2023222 
(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))))