Quotient of sum of exps

Percentage Accurate: 98.9% → 99.4%

Time: 4.7s

Alternatives: 6

Speedup: 0.5×

• 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: 98.9% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (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) :: t_0
    real(8) :: tmp
    t_0 = exp(a) / (exp(a) + exp(b))
    if (t_0 <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0 72.3%

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

4. Final simplification 99.2%

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

Alternative 2: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999965:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999965)
   (exp (- a (log1p (exp a))))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999965) {
		tmp = exp((a - log1p(exp(a))));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999999965) {
		tmp = Math.exp((a - Math.log1p(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) <= 0.999999965:
		tmp = math.exp((a - math.log1p(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) <= 0.999999965)
		tmp = exp(Float64(a - log1p(exp(a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999965], N[Exp[N[(a - N[Log[1 + N[Exp[a], $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) < 0.99999996499999999

   1. Initial program 94.2%

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

   2. Taylor expanded in b around 0 97.1%

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

      1. expm1-log1p-u 97.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{1 + e^{a}}\right)\right)} \]

      2. expm1-udef 97.1%

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

      3. add-exp-log 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{e^{a}}{1 + e^{a}}\right)}}\right)} - 1 \]

      4. log-div 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\log \left(e^{a}\right) - \log \left(1 + e^{a}\right)}}\right)} - 1 \]

      5. add-log-exp 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{a} - \log \left(1 + e^{a}\right)}\right)} - 1 \]

      6. log1p-udef 97.1%

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

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 97.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)\right)} \]

      2. expm1-log1p 97.2%

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

   6. Simplified 97.2%

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

   if 0.99999996499999999 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999965:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999965:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999965)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999965) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999999965d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999999965) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999999965:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999965)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999999965)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999965], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999996499999999

   1. Initial program 94.2%

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

   2. Taylor expanded in b around 0 97.1%

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

   if 0.99999996499999999 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999965:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4: 65.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

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

Julia

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

   1. Initial program 93.9%

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

   2. Taylor expanded in b around 0 97.1%

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

      1. expm1-log1p-u 97.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{1 + e^{a}}\right)\right)} \]

      2. expm1-udef 97.1%

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

      3. add-exp-log 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{e^{a}}{1 + e^{a}}\right)}}\right)} - 1 \]

      4. log-div 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\log \left(e^{a}\right) - \log \left(1 + e^{a}\right)}}\right)} - 1 \]

      5. add-log-exp 97.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{a} - \log \left(1 + e^{a}\right)}\right)} - 1 \]

      6. log1p-udef 97.1%

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

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 97.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)\right)} \]

      2. expm1-log1p 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in a around inf 97.1%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in b around 0 55.5%

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

   3. Taylor expanded in a around 0 54.5%

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

      1. *-commutative 54.5%

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

   5. Simplified 54.5%

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

4. Final simplification 65.5%

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

Alternative 5: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5600000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5600000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5600000.0) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5600000.0d0)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5600000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5600000.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5600000.0)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5600000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5600000.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.6e6

   1. Initial program 96.8%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{1 + e^{a}}\right)\right)} \]

      2. expm1-udef 100.0%

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

      3. add-exp-log 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{e^{a}}{1 + e^{a}}\right)}}\right)} - 1 \]

      4. log-div 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\log \left(e^{a}\right) - \log \left(1 + e^{a}\right)}}\right)} - 1 \]

      5. add-log-exp 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{a} - \log \left(1 + e^{a}\right)}\right)} - 1 \]

      6. log1p-udef 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a - \mathsf{log1p}\left(e^{a}\right)}\right)\right)} \]

      2. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -5.6e6 < a

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5600000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6: 39.6% 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 97.2%

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

2. Taylor expanded in a around 0 81.9%

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

3. Taylor expanded in b around 0 41.0%

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

4. Final simplification 41.0%

   \[\leadsto 0.5 \]

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 2023322 
(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))))