Quotient of sum of exps

Percentage Accurate: 98.7% → 100.0%

Time: 9.0s

Alternatives: 14

Speedup: 2.7×

• could not determine a ground truth

  1. a = 7.651187167493154e+121
  2. b = 170488.4559264945

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: 98.7% 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: 100.0% accurate, 2.7× 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]

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. rec-exp N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   9. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in a around inf

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

   1. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}}} \]

   2. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} \cdot e^{a} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}} \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{1}{\color{blue}{1} + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{1 + e^{\mathsf{neg}\left(a\right)} \cdot e^{b}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{1 + \color{blue}{e^{b} \cdot e^{\mathsf{neg}\left(a\right)}}} \]

   6. neg-mul-1 N/A

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

   7. prod-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 100.0

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

7. Applied rewrites 100.0%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{b - a}}} \]
8. Add Preprocessing

Alternative 2: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	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) <= 0.9999d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.9999) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9999:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.9999)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 3: 98.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -250000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -250000.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-250000.0d0)) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -250000.0:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -250000.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -250000.0)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -250000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -2.5e5 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 4: 92.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e+86)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.9e+86) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.9e+86)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.9e+86], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8999999999999999e86

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right)}, 2\right)} \]

   if -2.8999999999999999e86 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 93.8

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

   5. Applied rewrites 93.8%

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

Alternative 5: 85.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.8)
   (+ 1.0 (exp b))
   (if (<= b 2.8e+96)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.8) {
		tmp = 1.0 + exp(b);
	} else if (b <= 2.8e+96) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.8)
		tmp = Float64(1.0 + exp(b));
	elseif (b <= 2.8e+96)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.8], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e+96], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.79999999999999982

   1. Initial program 96.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if -6.79999999999999982 < b < 2.8e96

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 88.4

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

   7. Applied rewrites 88.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right)}, 2\right)} \]

   if 2.8e96 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)}, 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+96)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+96) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.8e+96)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.8e+96], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e96

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 70.6

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right)}, 2\right)} \]

   if 2.8e96 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+96)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+96) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.8e+96)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.8e+96], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e96

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 70.6

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{a \cdot \left(\frac{1}{2} \cdot a - 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, -1\right)}, 2\right)} \]

   if 2.8e96 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e+135)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e+135) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e+135)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e+135], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3e135

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 68.6

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

   7. Applied rewrites 68.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{a \cdot \left(\frac{1}{2} \cdot a - 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, -1\right)}, 2\right)} \]

   if 3e135 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0004:\\ \;\;\;\;\frac{1}{1 + \left(1 - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.0004)
   (/ 1.0 (+ 1.0 (- 1.0 a)))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0004) {
		tmp = 1.0 / (1.0 + (1.0 - a));
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0004)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 - a)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0004], N[(1.0 / N[(1.0 + N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000019e-4

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 76.4

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{-1 \cdot a}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.3%

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

   if 4.00000000000000019e-4 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -250000:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{1 + \left(1 + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -250000.0)
   (* 0.020833333333333332 (* b (* b b)))
   (/ (+ 1.0 a) (+ 1.0 (+ 1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = (1.0 + a) / (1.0 + (1.0 + a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-250000.0d0)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = (1.0d0 + a) / (1.0d0 + (1.0d0 + a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = (1.0 + a) / (1.0 + (1.0 + a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -250000.0:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = (1.0 + a) / (1.0 + (1.0 + a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -250000.0)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(1.0 + Float64(1.0 + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -250000.0)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = (1.0 + a) / (1.0 + (1.0 + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -250000.0], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(1.0 + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.6%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(0.020833333333333332, b \cdot b, -0.25\right)}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto {b}^{3} \cdot \left(\left(\frac{1}{48} + \frac{\frac{1}{2}}{{b}^{3}}\right) - \color{blue}{\frac{1}{4} \cdot \frac{1}{{b}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 1.8%

       \[\leadsto b \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(0.020833333333333332 - \frac{0.25}{b \cdot b}\right) + \frac{0.5}{b \cdot \left(b \cdot b\right)}\right)}\right) \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{48} \cdot {b}^{3} \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto 0.020833333333333332 \cdot \left(b \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

   if -2.5e5 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.7

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

   11. Applied rewrites 51.7%

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

   12. Taylor expanded in a around 0

       \[\leadsto \frac{1 + a}{\color{blue}{\left(1 + a\right)} + 1} \]
   13. Step-by-step derivation

       1. lower-+.f64 52.8

          \[\leadsto \frac{1 + a}{\color{blue}{\left(1 + a\right)} + 1} \]

   14. Applied rewrites 52.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -250000:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{1 + \left(1 + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -250000:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -250000.0)
   (* 0.020833333333333332 (* b (* b b)))
   (/ 1.0 (+ 1.0 (- 1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (1.0 + (1.0 - a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-250000.0d0)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (1.0d0 + (1.0d0 - a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (1.0 + (1.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -250000.0:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (1.0 + (1.0 - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -250000.0)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -250000.0)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (1.0 + (1.0 - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -250000.0], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.6%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(0.020833333333333332, b \cdot b, -0.25\right)}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto {b}^{3} \cdot \left(\left(\frac{1}{48} + \frac{\frac{1}{2}}{{b}^{3}}\right) - \color{blue}{\frac{1}{4} \cdot \frac{1}{{b}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 1.8%

       \[\leadsto b \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(0.020833333333333332 - \frac{0.25}{b \cdot b}\right) + \frac{0.5}{b \cdot \left(b \cdot b\right)}\right)}\right) \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{48} \cdot {b}^{3} \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto 0.020833333333333332 \cdot \left(b \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

   if -2.5e5 < a

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{-1 \cdot a}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.6%

       \[\leadsto \frac{1}{1 + \left(1 - \color{blue}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -250000:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -250000.0)
   (* 0.020833333333333332 (* b (* b b)))
   (/ 1.0 (- 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-250000.0d0)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -250000.0:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -250000.0)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -250000.0)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -250000.0], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.6%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(0.020833333333333332, b \cdot b, -0.25\right)}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto {b}^{3} \cdot \left(\left(\frac{1}{48} + \frac{\frac{1}{2}}{{b}^{3}}\right) - \color{blue}{\frac{1}{4} \cdot \frac{1}{{b}^{2}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 1.8%

       \[\leadsto b \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(0.020833333333333332 - \frac{0.25}{b \cdot b}\right) + \frac{0.5}{b \cdot \left(b \cdot b\right)}\right)}\right) \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{48} \cdot {b}^{3} \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto 0.020833333333333332 \cdot \left(b \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

   if -2.5e5 < a

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. rec-exp N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

      9. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.6%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. rec-exp N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot \color{blue}{e^{\mathsf{neg}\left(a\right)}}} \]

   9. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. exp-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lower-exp.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-neg.f64 64.0

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

7. Applied rewrites 64.0%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot a}} \]
9. Step-by-step derivation

10. Applied rewrites 42.8%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]
11. Add Preprocessing

Alternative 14: 39.7% accurate, 315.0× 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. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 85.3

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

5. Applied rewrites 85.3%

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

6. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 41.6%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× 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 2024219 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))