Quotient of sum of exps

Percentage Accurate: 99.1% → 99.1%

Time: 6.7s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

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 2: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-206}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-206) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-206) {
		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) <= 1d-206) 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) <= 1e-206) {
		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) <= 1e-206:
		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) <= 1e-206)
		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) <= 1e-206)
		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], 1e-206], N[Exp[a], $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000003e-206

   1. Initial program 100.0%

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

      1. add-cbrt-cube 98.7%

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

      2. pow1/3 98.7%

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

      3. pow-to-exp 98.7%

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

      4. pow3 98.7%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if 1.00000000000000003e-206 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-206}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3: 82.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 2500 \lor \neg \left(b \leq 1.45 \cdot 10^{+118}\right) \land b \leq 3 \cdot 10^{+202}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.1)
   (exp a)
   (if (or (<= b 2500.0) (and (not (<= b 1.45e+118)) (<= b 3e+202)))
     (/ (exp a) 2.0)
     (* -0.020833333333333332 (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = exp(a);
	} else if ((b <= 2500.0) || (!(b <= 1.45e+118) && (b <= 3e+202))) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = -0.020833333333333332 * pow(a, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = exp(a)
    else if ((b <= 2500.0d0) .or. (.not. (b <= 1.45d+118)) .and. (b <= 3d+202)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (-0.020833333333333332d0) * (a ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = Math.exp(a);
	} else if ((b <= 2500.0) || (!(b <= 1.45e+118) && (b <= 3e+202))) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = math.exp(a)
	elif (b <= 2500.0) or (not (b <= 1.45e+118) and (b <= 3e+202)):
		tmp = math.exp(a) / 2.0
	else:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = exp(a);
	elseif ((b <= 2500.0) || (!(b <= 1.45e+118) && (b <= 3e+202)))
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = exp(a);
	elseif ((b <= 2500.0) || (~((b <= 1.45e+118)) && (b <= 3e+202)))
		tmp = exp(a) / 2.0;
	else
		tmp = -0.020833333333333332 * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], N[Exp[a], $MachinePrecision], If[Or[LessEqual[b, 2500.0], And[N[Not[LessEqual[b, 1.45e+118]], $MachinePrecision], LessEqual[b, 3e+202]]], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 95.9%

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

   if -1.1000000000000001 < b < 2500 or 1.45000000000000008e118 < b < 3.0000000000000001e202

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0 91.9%

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

   3. Taylor expanded in a around 0 89.7%

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

   if 2500 < b < 1.45000000000000008e118 or 3.0000000000000001e202 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 25.7%

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

   3. Taylor expanded in a around 0 2.9%

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

   4. Taylor expanded in a around inf 55.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 2500 \lor \neg \left(b \leq 1.45 \cdot 10^{+118}\right) \land b \leq 3 \cdot 10^{+202}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \]

Alternative 4: 63.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-12} \lor \neg \left(a \leq -2.45 \cdot 10^{-30}\right) \land a \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.2e-12) (and (not (<= a -2.45e-30)) (<= a -6.5e-81)))
   (exp a)
   (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5.2e-12) || (!(a <= -2.45e-30) && (a <= -6.5e-81))) {
		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 ((a <= (-5.2d-12)) .or. (.not. (a <= (-2.45d-30))) .and. (a <= (-6.5d-81))) 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 ((a <= -5.2e-12) || (!(a <= -2.45e-30) && (a <= -6.5e-81))) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -5.2e-12) or (not (a <= -2.45e-30) and (a <= -6.5e-81)):
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5.2e-12) || (!(a <= -2.45e-30) && (a <= -6.5e-81)))
		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 ((a <= -5.2e-12) || (~((a <= -2.45e-30)) && (a <= -6.5e-81)))
		tmp = exp(a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5.2e-12], And[N[Not[LessEqual[a, -2.45e-30]], $MachinePrecision], LessEqual[a, -6.5e-81]]], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999965e-12 or -2.44999999999999985e-30 < a < -6.5000000000000002e-81

   1. Initial program 100.0%

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

      1. add-cbrt-cube 98.9%

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

      2. pow1/3 99.0%

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

      3. pow-to-exp 98.9%

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

      4. pow3 98.9%

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

      5. log-pow 99.9%

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

      6. log-div 99.9%

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

      7. add-log-exp 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in a around inf 93.9%

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

   if -5.19999999999999965e-12 < a < -2.44999999999999985e-30 or -6.5000000000000002e-81 < a

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0 52.0%

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

   3. Taylor expanded in a around 0 51.0%

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

      1. *-commutative 51.0%

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

   5. Simplified 51.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-12} \lor \neg \left(a \leq -2.45 \cdot 10^{-30}\right) \land a \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 5: 79.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = exp(a)
    else
        tmp = exp(a) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 95.9%

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

   if -1.1000000000000001 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 77.3%

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

   3. Taylor expanded in a around 0 75.6%

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

4. Final simplification 80.4%

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

Alternative 6: 39.4% 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 63.7%

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

3. Taylor expanded in a around 0 35.5%

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

   1. *-commutative 35.5%

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

5. Simplified 35.5%

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

6. Final simplification 35.5%

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

Alternative 7: 39.2% 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 b around 0 63.7%

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

3. Taylor expanded in a around 0 35.2%

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

4. Final simplification 35.2%

   \[\leadsto 0.5 \]

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 2023189 
(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))))