Quotient of sum of exps

Percentage Accurate: 98.8% → 99.1%

Time: 6.9s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% 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, 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 98.4%

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

   1. add-exp-log 98.4%

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

   2. div-exp 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99998)
   (/ (exp a) (+ (exp a) 1.0))
   (exp (- (log1p (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99998) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = exp(-log1p(exp(b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99998) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.exp(-Math.log1p(Math.exp(b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99998:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = math.exp(-math.log1p(math.exp(b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99998)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = exp(Float64(-log1p(exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99998], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99997999999999998

   1. Initial program 98.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

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

   if 0.99997999999999998 < (exp.f64 a)

   1. Initial program 98.3%

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

      1. add-exp-log 98.3%

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

      2. div-exp 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in a around 0 98.3%

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

      1. log1p-def 98.3%

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

      2. neg-mul-1 98.3%

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

   7. Simplified 98.3%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99998) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99998) {
		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) <= 0.99998d0) 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) <= 0.99998) {
		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) <= 0.99998:
		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) <= 0.99998)
		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) <= 0.99998)
		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], 0.99998], 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) < 0.99997999999999998

   1. Initial program 98.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

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

   if 0.99997999999999998 < (exp.f64 a)

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Final simplification 98.4%

   \[\leadsto \frac{e^{a}}{e^{a} + e^{b}} \]
4. Add Preprocessing

Alternative 5: 98.5% 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 98.7%

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

      1. add-exp-log 98.7%

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

      2. div-exp 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 97.6%

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

4. Final simplification 98.3%

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

Alternative 6: 65.3% 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 98.7%

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

      1. add-exp-log 98.7%

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

      2. div-exp 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 47.0%

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

   4. Taylor expanded in a around 0 46.1%

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

      1. *-commutative 46.1%

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

   6. Simplified 46.1%

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

4. Final simplification 62.5%

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

Alternative 7: 54.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -5.4e-8) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -5.4e-8) {
		tmp = 1.0;
	} 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 (b <= (-5.4d-8)) then
        tmp = 1.0d0
    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 (b <= -5.4e-8) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -5.4e-8:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -5.4e-8)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -5.4e-8)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -5.4e-8], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.40000000000000005e-8

   1. Initial program 98.1%

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

      1. add-exp-log 98.1%

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

      2. div-exp 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in a around 0 96.3%

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

      1. log1p-def 96.3%

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

      2. neg-mul-1 96.3%

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

   7. Simplified 96.3%

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

      1. add-sqr-sqrt 96.3%

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

      2. pow1/2 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5}} \]

      3. neg-mul-1 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{b}\right)}}\right)}^{0.5} \]

      4. *-commutative 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\mathsf{log1p}\left(e^{b}\right) \cdot -1}}\right)}^{0.5} \]

      5. add-sqr-sqrt 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      6. sqrt-unprod 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1}\right)}^{0.5} \]

      7. sqr-neg 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1}\right)}^{0.5} \]

      8. sqrt-unprod 96.2%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      9. add-sqr-sqrt 96.3%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1}\right)}^{0.5} \]

      10. add-log-exp 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      11. pow-to-exp 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)}}^{0.5} \]

      12. pow-pow 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      13. metadata-eval 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{-0.5}} \]

      14. metadata-eval 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{\left(0.5 + -1\right)}} \]

      15. pow-prod-up 96.3%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)} \]

   9. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{1} \]

   if -5.40000000000000005e-8 < b

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 73.7%

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

   4. Taylor expanded in a around 0 36.5%

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

      1. *-commutative 36.5%

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

   6. Simplified 36.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -170000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -170000.0) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -170000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 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 <= (-170000.0d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -170000.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -170000.0:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -170000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -170000.0)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -170000.0], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.7e5

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow1/2 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5}} \]

      3. neg-mul-1 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{b}\right)}}\right)}^{0.5} \]

      4. *-commutative 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\mathsf{log1p}\left(e^{b}\right) \cdot -1}}\right)}^{0.5} \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      6. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1}\right)}^{0.5} \]

      7. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1}\right)}^{0.5} \]

      8. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1}\right)}^{0.5} \]

      10. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      11. pow-to-exp 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)}}^{0.5} \]

      12. pow-pow 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      13. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{-0.5}} \]

      14. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{\left(0.5 + -1\right)}} \]

      15. pow-prod-up 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.7e5 < b

   1. Initial program 98.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 73.9%

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

   4. Taylor expanded in b around 0 37.1%

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

      1. +-commutative 37.1%

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

   6. Simplified 37.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -170000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.8% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -170000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -170000.0) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -170000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-170000.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -170000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -170000.0:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -170000.0)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -170000.0)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -170000.0], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -1.7e5

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow1/2 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5}} \]

      3. neg-mul-1 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{-1 \cdot \mathsf{log1p}\left(e^{b}\right)}}\right)}^{0.5} \]

      4. *-commutative 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\mathsf{log1p}\left(e^{b}\right) \cdot -1}}\right)}^{0.5} \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      6. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1}\right)}^{0.5} \]

      7. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1}\right)}^{0.5} \]

      8. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      9. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1}\right)}^{0.5} \]

      10. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1}\right)}^{0.5} \]

      11. pow-to-exp 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)}}^{0.5} \]

      12. pow-pow 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      13. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{-0.5}} \]

      14. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{\left(0.5 + -1\right)}} \]

      15. pow-prod-up 100.0%

          \[\leadsto \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \color{blue}{\left({\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{0.5} \cdot {\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}\right)} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1} \]

   if -1.7e5 < b

   1. Initial program 98.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 73.9%

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

   4. Taylor expanded in b around 0 35.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -170000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.1% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 79.0%

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

4. Taylor expanded in b around 0 32.4%

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

5. Final simplification 32.4%

   \[\leadsto 0.5 \]
6. Add Preprocessing

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 2024020 
(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))))