Quotient of sum of exps

Percentage Accurate: 98.8% → 98.8%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 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 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999999998)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999999998) {
		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) <= 0.999999999998d0) 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) <= 0.999999999998) {
		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) <= 0.999999999998:
		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) <= 0.999999999998)
		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) <= 0.999999999998)
		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], 0.999999999998], 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) < 0.99999999999800004

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 98.8%

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

   if 0.99999999999800004 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999998:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3: 64.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -480:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-300} \lor \neg \left(a \leq 3.6 \cdot 10^{-286}\right):\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -480.0)
   (/ (exp a) a)
   (if (or (<= a -2.1e-300) (not (<= a 3.6e-286)))
     (+ 0.5 (* a (+ 0.25 (* a 0.125))))
     (* -0.020833333333333332 (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -480.0) {
		tmp = exp(a) / a;
	} else if ((a <= -2.1e-300) || !(a <= 3.6e-286)) {
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	} 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 (a <= (-480.0d0)) then
        tmp = exp(a) / a
    else if ((a <= (-2.1d-300)) .or. (.not. (a <= 3.6d-286))) then
        tmp = 0.5d0 + (a * (0.25d0 + (a * 0.125d0)))
    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 (a <= -480.0) {
		tmp = Math.exp(a) / a;
	} else if ((a <= -2.1e-300) || !(a <= 3.6e-286)) {
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	} else {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -480.0:
		tmp = math.exp(a) / a
	elif (a <= -2.1e-300) or not (a <= 3.6e-286):
		tmp = 0.5 + (a * (0.25 + (a * 0.125)))
	else:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -480.0)
		tmp = Float64(exp(a) / a);
	elseif ((a <= -2.1e-300) || !(a <= 3.6e-286))
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(a * 0.125))));
	else
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -480.0)
		tmp = exp(a) / a;
	elseif ((a <= -2.1e-300) || ~((a <= 3.6e-286)))
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	else
		tmp = -0.020833333333333332 * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -480.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], If[Or[LessEqual[a, -2.1e-300], N[Not[LessEqual[a, 3.6e-286]], $MachinePrecision]], N[(0.5 + N[(a * N[(0.25 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -480

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -480 < a < -2.10000000000000004e-300 or 3.60000000000000013e-286 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 59.8%

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

   3. Taylor expanded in a around 0 59.2%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in a around 0 59.3%

      \[\leadsto \color{blue}{0.5 + \left(0.125 \cdot {a}^{2} + 0.25 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative 59.3%

         \[\leadsto 0.5 + \color{blue}{\left(0.25 \cdot a + 0.125 \cdot {a}^{2}\right)} \]

      2. unpow2 59.3%

         \[\leadsto 0.5 + \left(0.25 \cdot a + 0.125 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      3. associate-*r* 59.3%

         \[\leadsto 0.5 + \left(0.25 \cdot a + \color{blue}{\left(0.125 \cdot a\right) \cdot a}\right) \]

      4. distribute-rgt-out 59.3%

         \[\leadsto 0.5 + \color{blue}{a \cdot \left(0.25 + 0.125 \cdot a\right)} \]

   8. Simplified 59.3%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + 0.125 \cdot a\right)} \]

   if -2.10000000000000004e-300 < a < 3.60000000000000013e-286

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 27.9%

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

   3. Taylor expanded in a around 0 27.9%

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

   4. Taylor expanded in a around inf 62.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -480:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -2.1 \cdot 10^{-300} \lor \neg \left(a \leq 3.6 \cdot 10^{-286}\right):\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \]

Alternative 4: 97.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+24}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e+24) (/ (exp a) a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.9e+24) {
		tmp = exp(a) / 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 <= (-2.9d+24)) then
        tmp = exp(a) / 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 <= -2.9e+24) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.9e+24:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.9e+24)
		tmp = Float64(exp(a) / 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 <= -2.9e+24)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.9e+24], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999979e24

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -2.89999999999999979e24 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+24}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 5: 97.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.55e-12) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.55e-12) {
		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 (a <= (-3.55d-12)) 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 (a <= -3.55e-12) {
		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 a <= -3.55e-12:
		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 (a <= -3.55e-12)
		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 (a <= -3.55e-12)
		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[a, -3.55e-12], 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 a < -3.55e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 98.8%

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

   3. Taylor expanded in a around 0 97.2%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 97.2%

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

   5. Simplified 97.2%

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

   if -3.55e-12 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 99.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6: 65.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -420:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -420.0) (/ (exp a) a) (+ 0.5 (* a (+ 0.25 (* a 0.125))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -420.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-420.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 0.5d0 + (a * (0.25d0 + (a * 0.125d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -420.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -420.0:
		tmp = math.exp(a) / a
	else:
		tmp = 0.5 + (a * (0.25 + (a * 0.125)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -420.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -420.0)
		tmp = exp(a) / a;
	else
		tmp = 0.5 + (a * (0.25 + (a * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -420.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(0.5 + N[(a * N[(0.25 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -420

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -420 < a

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 57.6%

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

   3. Taylor expanded in a around 0 57.1%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 57.1%

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

   5. Simplified 57.1%

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

   6. Taylor expanded in a around 0 57.1%

      \[\leadsto \color{blue}{0.5 + \left(0.125 \cdot {a}^{2} + 0.25 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative 57.1%

         \[\leadsto 0.5 + \color{blue}{\left(0.25 \cdot a + 0.125 \cdot {a}^{2}\right)} \]

      2. unpow2 57.1%

         \[\leadsto 0.5 + \left(0.25 \cdot a + 0.125 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      3. associate-*r* 57.1%

         \[\leadsto 0.5 + \left(0.25 \cdot a + \color{blue}{\left(0.125 \cdot a\right) \cdot a}\right) \]

      4. distribute-rgt-out 57.1%

         \[\leadsto 0.5 + \color{blue}{a \cdot \left(0.25 + 0.125 \cdot a\right)} \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + 0.125 \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -420:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot 0.125\right)\\ \end{array} \]

Alternative 7: 39.8% 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 100.0%

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

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 43.6%

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

   1. *-commutative 43.6%

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

5. Simplified 43.6%

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

6. Final simplification 43.6%

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

Alternative 8: 39.7% 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 100.0%

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

2. Taylor expanded in a around 0 81.9%

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

3. Taylor expanded in b around 0 43.2%

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

4. Final simplification 43.2%

   \[\leadsto 0.5 \]

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 2023330 
(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))))