Quotient of sum of exps

Percentage Accurate: 99.0% → 99.1%

Time: 8.4s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.2593647209070981e+122
  2. b = -1.2912342823817406e+211

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (fma (log (+ (exp a) (exp b))) -1.0 a)))

C

double code(double a, double b) {
	return exp(fma(log((exp(a) + exp(b))), -1.0, a));
}

Julia

function code(a, b)
	return exp(fma(log(Float64(exp(a) + exp(b))), -1.0, a))
end

Wolfram

code[a_, b_] := N[Exp[N[(N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0 + a), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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. associate-/r/ N/A

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

   4. inv-pow N/A

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

   5. pow-to-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

   10. lower-log.f64 98.8

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
5. Add Preprocessing

Alternative 2: 68.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b \cdot b, b\right)}\\ \mathbf{elif}\;t\_0 \leq 0.505506018332186:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(0.020833333333333332, b \cdot b, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 0.002)
     (/ 1.0 (fma 0.5 (* b b) b))
     (if (<= t_0 0.505506018332186)
       (fma b (fma 0.020833333333333332 (* b b) -0.25) 0.5)
       1.0))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = 1.0 / fma(0.5, (b * b), b);
	} else if (t_0 <= 0.505506018332186) {
		tmp = fma(b, fma(0.020833333333333332, (b * b), -0.25), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(1.0 / fma(0.5, Float64(b * b), b));
	elseif (t_0 <= 0.505506018332186)
		tmp = fma(b, fma(0.020833333333333332, Float64(b * b), -0.25), 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(1.0 / N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.505506018332186], N[(b * N[(0.020833333333333332 * N[(b * b), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 2e-3

   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 64.9

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

   5. Applied rewrites 64.9%

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

   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 34.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 33.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, b \cdot \color{blue}{b}, b\right)} \]

   if 2e-3 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 97.0%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 3: 68.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;t\_0 \leq 0.505506018332186:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(0.020833333333333332, b \cdot b, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 0.002)
     (/ 1.0 (* 0.5 (* b b)))
     (if (<= t_0 0.505506018332186)
       (fma b (fma 0.020833333333333332 (* b b) -0.25) 0.5)
       1.0))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = 1.0 / (0.5 * (b * b));
	} else if (t_0 <= 0.505506018332186) {
		tmp = fma(b, fma(0.020833333333333332, (b * b), -0.25), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(1.0 / Float64(0.5 * Float64(b * b)));
	elseif (t_0 <= 0.505506018332186)
		tmp = fma(b, fma(0.020833333333333332, Float64(b * b), -0.25), 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(1.0 / N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.505506018332186], N[(b * N[(0.020833333333333332 * N[(b * b), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 2e-3

   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 64.9

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

   5. Applied rewrites 64.9%

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

   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 34.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 33.5%

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

   if 2e-3 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 97.0%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 4: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\frac{64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}{\mathsf{fma}\left(b, -16, 32\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (/ (- 64.0 (* (* b b) (* (* b b) (* b b)))) (fma b -16.0 32.0)))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / ((64.0 - ((b * b) * ((b * b) * (b * b)))) / fma(b, -16.0, 32.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / Float64(Float64(64.0 - Float64(Float64(b * b) * Float64(Float64(b * b) * Float64(b * b)))) / fma(b, -16.0, 32.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(N[(64.0 - N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * -16.0 + 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.6%

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

   10. Applied rewrites 42.4%

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{\left(8 - b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b - 2, 4\right)}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{32 + -16 \cdot \color{blue}{b}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.4%

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{\mathsf{fma}\left(b, -16, 32\right)}} \]

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\frac{64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}{\mathsf{fma}\left(b, -16, 32\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\frac{64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}{32}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (/ (- 64.0 (* (* b b) (* (* b b) (* b b)))) 32.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / ((64.0 - ((b * b) * ((b * b) * (b * b)))) / 32.0);
	} else {
		tmp = 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) / (exp(a) + exp(b))) <= 0.505506018332186d0) then
        tmp = 1.0d0 / ((64.0d0 - ((b * b) * ((b * b) * (b * b)))) / 32.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(a) + Math.exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / ((64.0 - ((b * b) * ((b * b) * (b * b)))) / 32.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(a) + math.exp(b))) <= 0.505506018332186:
		tmp = 1.0 / ((64.0 - ((b * b) * ((b * b) * (b * b)))) / 32.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / Float64(Float64(64.0 - Float64(Float64(b * b) * Float64(Float64(b * b) * Float64(b * b)))) / 32.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186)
		tmp = 1.0 / ((64.0 - ((b * b) * ((b * b) * (b * b)))) / 32.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(N[(64.0 - N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 32.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.6%

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

   10. Applied rewrites 42.4%

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{\left(8 - b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b - 2, 4\right)}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{32}} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.0%

       \[\leadsto \frac{1}{\frac{\left(64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot 1}{32}} \]

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\frac{64 - \left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}{32}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   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 66.2%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 7: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (fma b (fma 0.5 b 1.0) 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / fma(b, fma(0.5, b, 1.0), 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / fma(b, fma(0.5, b, 1.0), 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(b * N[(0.5 * b + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   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 59.9%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 8: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot 0.5, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (fma b (* b 0.5) 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / fma(b, (b * 0.5), 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / fma(b, Float64(b * 0.5), 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   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 59.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \frac{1}{2} \cdot b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, 0.5 \cdot b, 2\right)} \]

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot 0.5, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (/ 1.0 (+ b 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 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) / (exp(a) + exp(b))) <= 0.505506018332186d0) then
        tmp = 1.0d0 / (b + 2.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(a) + Math.exp(b))) <= 0.505506018332186) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(a) + math.exp(b))) <= 0.505506018332186:
		tmp = 1.0 / (b + 2.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = Float64(1.0 / Float64(b + 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186)
		tmp = 1.0 / (b + 2.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.6%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;\mathsf{fma}\left(b, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186)
   (fma b -0.25 0.5)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = fma(b, -0.25, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = fma(b, -0.25, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], N[(b * -0.25 + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot b} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

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

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 11: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.505506018332186:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.505506018332186) 0.5 1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186) {
		tmp = 0.5;
	} else {
		tmp = 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) / (exp(a) + exp(b))) <= 0.505506018332186d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(a) + Math.exp(b))) <= 0.505506018332186) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(a) + math.exp(b))) <= 0.505506018332186:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.505506018332186)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.505506018332186)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.505506018332186], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.505506018332186

   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 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 41.5%

      \[\leadsto 0.5 \]

   if 0.505506018332186 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      10. lower-log.f64 95.0

          \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{{\log \left(e^{a} + e^{b}\right)}^{2} - a \cdot a}\right)}^{\left(\frac{1}{\left(-\log \left(e^{a} + e^{b}\right)\right) - a}\right)}} \]

   7. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

Alternative 12: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 13: 98.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -480000000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -480000000.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -480000000.0) {
		tmp = exp(a) / (1.0 + 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 (a <= (-480000000.0d0)) then
        tmp = exp(a) / (1.0d0 + 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 (a <= -480000000.0) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -480000000.0:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -480000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + 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 (a <= -480000000.0)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -480000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 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 a < -4.8e8

   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 -4.8e8 < a

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -480000000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3500000000:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3500000000.0)
   (* -0.0020833333333333333 (pow b 5.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3500000000.0) {
		tmp = -0.0020833333333333333 * pow(b, 5.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3500000000.0d0)) then
        tmp = (-0.0020833333333333333d0) * (b ** 5.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3500000000.0) {
		tmp = -0.0020833333333333333 * Math.pow(b, 5.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3500000000.0:
		tmp = -0.0020833333333333333 * math.pow(b, 5.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3500000000.0)
		tmp = Float64(-0.0020833333333333333 * (b ^ 5.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3500000000.0)
		tmp = -0.0020833333333333333 * (b ^ 5.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3500000000.0], N[(-0.0020833333333333333 * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5e9

   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 26.0

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

   5. Applied rewrites 26.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.8%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right)}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.2%

       \[\leadsto -0.0020833333333333333 \cdot {b}^{\color{blue}{5}} \]

   if -3.5e9 < a

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3500000000:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 82.6

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

5. Applied rewrites 82.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.4%

   \[\leadsto 0.5 \]
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 2024223 
(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))))