Quotient of sum of exps

Percentage Accurate: 98.9% → 98.5%

Time: 5.5s

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: 98.9% 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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.01) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.01) {
		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) <= 0.01d0) 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) <= 0.01) {
		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) <= 0.01:
		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) <= 0.01)
		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) <= 0.01)
		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], 0.01], 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) < 0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   if 0.0100000000000000002 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 2: 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]

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 3: 69.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-144} \lor \neg \left(a \leq -6.8 \cdot 10^{-201}\right):\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0)
   (/ (exp a) a)
   (if (or (<= a -3.3e-144) (not (<= a -6.8e-201)))
     (/ 1.0 (+ 2.0 (+ b (* (* b b) 0.5))))
     (* -0.020833333333333332 (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = exp(a) / a;
	} else if ((a <= -3.3e-144) || !(a <= -6.8e-201)) {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	} 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 <= (-700.0d0)) then
        tmp = exp(a) / a
    else if ((a <= (-3.3d-144)) .or. (.not. (a <= (-6.8d-201)))) then
        tmp = 1.0d0 / (2.0d0 + (b + ((b * b) * 0.5d0)))
    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 <= -700.0) {
		tmp = Math.exp(a) / a;
	} else if ((a <= -3.3e-144) || !(a <= -6.8e-201)) {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	} else {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -700.0:
		tmp = math.exp(a) / a
	elif (a <= -3.3e-144) or not (a <= -6.8e-201):
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)))
	else:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -700.0)
		tmp = Float64(exp(a) / a);
	elseif ((a <= -3.3e-144) || !(a <= -6.8e-201))
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(Float64(b * b) * 0.5))));
	else
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -700.0)
		tmp = exp(a) / a;
	elseif ((a <= -3.3e-144) || ~((a <= -6.8e-201)))
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	else
		tmp = -0.020833333333333332 * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -700.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], If[Or[LessEqual[a, -3.3e-144], N[Not[LessEqual[a, -6.8e-201]], $MachinePrecision]], N[(1.0 / N[(2.0 + N[(b + N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -700

   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. Taylor expanded in a around inf 100.0%

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

   if -700 < a < -3.29999999999999995e-144 or -6.7999999999999997e-201 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.0%

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

   3. Taylor expanded in b around 0 53.5%

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

      1. *-commutative 53.5%

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

      2. unpow2 53.5%

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

   5. Simplified 53.5%

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

   if -3.29999999999999995e-144 < a < -6.7999999999999997e-201

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 7.4%

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

   3. Taylor expanded in a around 0 7.4%

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

   4. Taylor expanded in a around inf 73.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-144} \lor \neg \left(a \leq -6.8 \cdot 10^{-201}\right):\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \end{array} \]

Alternative 4: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3300:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3300.0) (/ (exp a) a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3300.0) {
		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 <= (-3300.0d0)) 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 <= -3300.0) {
		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 <= -3300.0:
		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 <= -3300.0)
		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 <= -3300.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3300.0], 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 < -3300

   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. Taylor expanded in a around inf 100.0%

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

   if -3300 < a

   1. Initial program 98.9%

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

   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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3300:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 5: 71.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0) (/ (exp a) a) (/ 1.0 (+ 2.0 (+ b (* (* b b) 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 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 (a <= (-700.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (2.0d0 + (b + ((b * b) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -700.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -700.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(Float64(b * b) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -700.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -700.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -700

   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. Taylor expanded in a around inf 100.0%

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

   if -700 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.1%

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

   3. Taylor expanded in b around 0 52.2%

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

      1. *-commutative 52.2%

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

      2. unpow2 52.2%

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

   5. Simplified 52.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 6: 52.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-114}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e-114) (+ 0.5 (* a 0.25)) (/ 1.0 (+ 2.0 (+ b (* (* b b) 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-114) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 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 <= 7.5d-114) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b + ((b * b) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-114) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.5e-114:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.5e-114)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(Float64(b * b) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.5e-114)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.5e-114], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5000000000000002e-114

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 68.4%

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

   3. Taylor expanded in a around 0 47.0%

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

   if 7.5000000000000002e-114 < b

   1. Initial program 98.8%

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

   2. Taylor expanded in a around 0 93.4%

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

   3. Taylor expanded in b around 0 55.2%

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

      1. *-commutative 55.2%

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

      2. unpow2 55.2%

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

   5. Simplified 55.2%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-114}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 7: 52.5% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-114}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-114) (+ 0.5 (* a 0.25)) (/ 1.0 (+ 2.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-114) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b * (b * 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 <= 1.1d-114) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-114) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1e-114:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (2.0 + (b * (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-114)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1e-114)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-114], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000006e-114

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 68.4%

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

   3. Taylor expanded in a around 0 47.0%

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

   if 1.10000000000000006e-114 < b

   1. Initial program 98.8%

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

   2. Taylor expanded in a around 0 93.4%

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

   3. Taylor expanded in b around 0 55.2%

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

      1. *-commutative 55.2%

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

      2. unpow2 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in b around inf 55.2%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 55.2%

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

      2. *-commutative 55.2%

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

      3. associate-*r* 55.2%

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

      4. *-commutative 55.2%

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

   8. Simplified 55.2%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-114}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]

Alternative 8: 52.6% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.4d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.4:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.4)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 70.7%

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

   3. Taylor expanded in a around 0 47.9%

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

   if 1.3999999999999999 < b

   1. Initial program 98.6%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 54.2%

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

      1. *-commutative 54.2%

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

      2. unpow2 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in b around inf 54.2%

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

      1. unpow2 54.2%

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

   8. Simplified 54.2%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 9: 38.8% 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 99.2%

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

2. Taylor expanded in b around 0 61.3%

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

3. Taylor expanded in a around 0 35.6%

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

4. Final simplification 35.6%

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

Alternative 10: 38.6% 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 99.2%

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

2. Taylor expanded in b around 0 61.3%

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

3. Taylor expanded in a around 0 35.4%

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

4. Final simplification 35.4%

   \[\leadsto 0.5 \]

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 2023181 
(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))))