Quotient of sum of exps

Percentage Accurate: 98.7% → 100.0%

Time: 9.6s

Alternatives: 15

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-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. 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-rgt-in N/A

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

   2. exp-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. lower-+.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. prod-exp N/A

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

   7. lower-exp.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. unsub-neg N/A

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

   10. 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: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6437559377674459:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.6437559377674459)
   (/
    1.0
    (fma
     b
     (fma b (/ (fma (* b (* b b)) 0.004629629629629629 0.125) 0.25) 1.0)
     2.0))
   (+ 1.0 (exp b))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.6437559377674459) {
		tmp = 1.0 / fma(b, fma(b, (fma((b * (b * b)), 0.004629629629629629, 0.125) / 0.25), 1.0), 2.0);
	} else {
		tmp = 1.0 + exp(b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.6437559377674459)
		tmp = Float64(1.0 / fma(b, fma(b, Float64(fma(Float64(b * Float64(b * b)), 0.004629629629629629, 0.125) / 0.25), 1.0), 2.0));
	else
		tmp = Float64(1.0 + exp(b));
	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], 0.6437559377674459], N[(1.0 / N[(b * N[(b * N[(N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.643755937767445885

   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 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 70.1

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

   8. Applied rewrites 70.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{{\left(b \cdot \frac{1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{{\left(b \cdot \frac{1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      3. unpow-prod-down N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\color{blue}{{b}^{3} \cdot {\frac{1}{6}}^{3}} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \color{blue}{\left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \color{blue}{\frac{1}{216}}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \color{blue}{\frac{1}{8}}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      10. associate-+r- N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, 1\right), 2\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, 1\right), 2\right)} \]

      12. swap-sqr N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\left(\color{blue}{\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)} + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\mathsf{fma}\left(b \cdot b, \frac{1}{6} \cdot \frac{1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right)} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{6} \cdot \frac{1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \color{blue}{\frac{1}{36}}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \color{blue}{\frac{1}{4}}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      19. metadata-eval 51.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, 0.25\right) - b \cdot \color{blue}{0.08333333333333333}}, 1\right), 2\right)} \]

   10. Applied rewrites 51.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, 0.25\right) - b \cdot 0.08333333333333333}}, 1\right), 2\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, 1\right), 2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 2\right)} \]

   if 0.643755937767445885 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   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%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6437559377674459:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.495:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{0.25}, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.495)
   (/
    1.0
    (fma
     b
     (fma b (/ (fma (* b (* b b)) 0.004629629629629629 0.125) 0.25) 1.0)
     2.0))
   (fma
    a
    (fma
     (* a a)
     (fma (* a a) 0.0020833333333333333 -0.020833333333333332)
     0.25)
    0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.495) {
		tmp = 1.0 / fma(b, fma(b, (fma((b * (b * b)), 0.004629629629629629, 0.125) / 0.25), 1.0), 2.0);
	} else {
		tmp = fma(a, fma((a * a), fma((a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.495)
		tmp = Float64(1.0 / fma(b, fma(b, Float64(fma(Float64(b * Float64(b * b)), 0.004629629629629629, 0.125) / 0.25), 1.0), 2.0));
	else
		tmp = fma(a, fma(Float64(a * a), fma(Float64(a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	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], 0.495], N[(1.0 / N[(b * N[(b * N[(N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333 + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.495

   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 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.0

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

   8. Applied rewrites 46.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{{\left(b \cdot \frac{1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{{\left(b \cdot \frac{1}{6}\right)}^{3} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      3. unpow-prod-down N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\color{blue}{{b}^{3} \cdot {\frac{1}{6}}^{3}} + {\frac{1}{2}}^{3}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\color{blue}{b \cdot \left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \color{blue}{\left(b \cdot b\right)}, {\frac{1}{6}}^{3}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \color{blue}{\frac{1}{216}}, {\frac{1}{2}}^{3}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \color{blue}{\frac{1}{8}}\right)}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}\right)}, 1\right), 2\right)} \]

      10. associate-+r- N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, 1\right), 2\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right) + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}}, 1\right), 2\right)} \]

      12. swap-sqr N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\left(\color{blue}{\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)} + \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\mathsf{fma}\left(b \cdot b, \frac{1}{6} \cdot \frac{1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right)} - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{6} \cdot \frac{1}{6}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \color{blue}{\frac{1}{36}}, \frac{1}{2} \cdot \frac{1}{2}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \color{blue}{\frac{1}{4}}\right) - \left(b \cdot \frac{1}{6}\right) \cdot \frac{1}{2}}, 1\right), 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{1}{4}\right) - \color{blue}{b \cdot \left(\frac{1}{6} \cdot \frac{1}{2}\right)}}, 1\right), 2\right)} \]

      19. metadata-eval 9.8

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, 0.25\right) - b \cdot \color{blue}{0.08333333333333333}}, 1\right), 2\right)} \]

   10. Applied rewrites 9.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, 0.25\right) - b \cdot 0.08333333333333333}}, 1\right), 2\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), \frac{1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, 1\right), 2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, 1\right), 2\right)} \]

   if 0.495 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, {a}^{2} \cdot \frac{1}{480} + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480}, \frac{-1}{48}\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.495:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.495)
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))
   (fma
    a
    (fma
     (* a a)
     (fma (* a a) 0.0020833333333333333 -0.020833333333333332)
     0.25)
    0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.495) {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	} else {
		tmp = fma(a, fma((a * a), fma((a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.495)
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	else
		tmp = fma(a, fma(Float64(a * a), fma(Float64(a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	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], 0.495], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333 + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.495

   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 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.0

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

   8. Applied rewrites 46.0%

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

   if 0.495 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, {a}^{2} \cdot \frac{1}{480} + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480}, \frac{-1}{48}\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 70.0

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

   8. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (fma b (* b (fma b 0.16666666666666666 0.5)) b))
   (fma
    a
    (fma
     (* a a)
     (fma (* a a) 0.0020833333333333333 -0.020833333333333332)
     0.25)
    0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma(b, (b * fma(b, 0.16666666666666666, 0.5)), b);
	} else {
		tmp = fma(a, fma((a * a), fma((a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), b));
	else
		tmp = fma(a, fma(Float64(a * a), fma(Float64(a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	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], 0.0], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333 + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\color{blue}{{b}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)}} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \frac{1}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{\left(b \cdot \color{blue}{{b}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)\right)}} \]

       4. associate-+r+ N/A

          \[\leadsto \frac{1}{b \cdot \left({b}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + \frac{1}{{b}^{2}}\right)}\right)} \]

       5. distribute-lft-in N/A

          \[\leadsto \frac{1}{b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)} + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       8. +-commutative N/A

          \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{6}\right)}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       9. distribute-rgt-in N/A

          \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b + \frac{1}{6} \cdot b\right)} + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right)} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       11. lft-mult-inverse N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       13. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + \color{blue}{1}\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{1}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + b \cdot 1}} \]

   11. Applied rewrites 45.9%

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

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, {a}^{2} \cdot \frac{1}{480} + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480}, \frac{-1}{48}\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (fma b (* b (fma b 0.16666666666666666 0.5)) b))
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma(b, (b * fma(b, 0.16666666666666666, 0.5)), b);
	} else {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), b));
	else
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	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], 0.0], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\color{blue}{{b}^{3} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)}} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \frac{1}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{\left(b \cdot \color{blue}{{b}^{2}}\right) \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)\right)\right)}} \]

       4. associate-+r+ N/A

          \[\leadsto \frac{1}{b \cdot \left({b}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + \frac{1}{{b}^{2}}\right)}\right)} \]

       5. distribute-lft-in N/A

          \[\leadsto \frac{1}{b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)} + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       8. +-commutative N/A

          \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{6}\right)}\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       9. distribute-rgt-in N/A

          \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b + \frac{1}{6} \cdot b\right)} + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right)} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       11. lft-mult-inverse N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot b\right) + {b}^{2} \cdot \frac{1}{{b}^{2}}\right)} \]

       13. rgt-mult-inverse N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + \color{blue}{1}\right)} \]

       14. distribute-lft-in N/A

           \[\leadsto \frac{1}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + b \cdot 1}} \]

   11. Applied rewrites 45.9%

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

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (* b (* b (fma b 0.16666666666666666 0.5))))
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / (b * (b * fma(b, 0.16666666666666666, 0.5)));
	} else {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))));
	else
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	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], 0.0], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\color{blue}{{b}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(b \cdot b\right) \cdot b\right)} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{{b}^{2}} \cdot b\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{2} \cdot \left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}} \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(b \cdot b\right)} \cdot \left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{6}\right)}\right)} \]

       6. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b + \frac{1}{6} \cdot b\right)}} \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right)} + \frac{1}{6} \cdot b\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{1} + \frac{1}{6} \cdot b\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{6} \cdot b\right)} \]

       10. associate-*r* N/A

           \[\leadsto \frac{1}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{1}{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]

       13. +-commutative N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)}\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{1}{b \cdot \left(b \cdot \left(\color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}\right)\right)} \]

       15. lower-fma.f64 45.8

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

   11. Applied rewrites 45.8%

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

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 6.0 (* b (* b b)))
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 6.0 / (b * (b * b));
	} else {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	else
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	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], 0.0], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]

       2. cube-mult N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot \left(b \cdot b\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{{b}^{2}}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot {b}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

       6. lower-*.f64 45.7

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

   11. Applied rewrites 45.7%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 2.0 (* b b))
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 2.0 / (b * b);
	} else {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(2.0 / Float64(b * b));
	else
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	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], 0.0], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 39.4

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

   8. Applied rewrites 39.4%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 39.1

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

   11. Applied rewrites 39.1%

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

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 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-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. 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 11: 98.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (a <= -250000.0) {
		tmp = exp(a) / b;
	} 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) / b
    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) / b;
	} 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) / b
	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) / b);
	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) / b;
	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] / b), $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

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   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 12: 59.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot b, \left(b \cdot b\right) \cdot 0.25, -4\right)} \cdot \mathsf{fma}\left(b, b \cdot 0.5, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e-5)
   (fma
    a
    (fma
     (* a a)
     (fma (* a a) 0.0020833333333333333 -0.020833333333333332)
     0.25)
    0.5)
   (if (<= b 2e+154)
     (* (/ 1.0 (fma (* b b) (* (* b b) 0.25) -4.0)) (fma b (* b 0.5) -2.0))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-5) {
		tmp = fma(a, fma((a * a), fma((a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	} else if (b <= 2e+154) {
		tmp = (1.0 / fma((b * b), ((b * b) * 0.25), -4.0)) * fma(b, (b * 0.5), -2.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.5e-5)
		tmp = fma(a, fma(Float64(a * a), fma(Float64(a * a), 0.0020833333333333333, -0.020833333333333332), 0.25), 0.5);
	elseif (b <= 2e+154)
		tmp = Float64(Float64(1.0 / fma(Float64(b * b), Float64(Float64(b * b) * 0.25), -4.0)) * fma(b, Float64(b * 0.5), -2.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.5e-5], N[(a * N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333 + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 2e+154], N[(N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * 0.5), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.49999999999999934e-5

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 75.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, {a}^{2} \cdot \frac{1}{480} + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480}, \frac{-1}{48}\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 57.6

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

   8. Applied rewrites 57.6%

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

   if 7.49999999999999934e-5 < b < 2.00000000000000007e154

   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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.2

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

   8. Applied rewrites 6.2%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 5.6

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

   11. Applied rewrites 5.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot 0.5}, 2\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. flip-+ N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

   13. Applied rewrites 54.0%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(b \cdot b, 0.25 \cdot \left(b \cdot b\right), -4\right)} \cdot \mathsf{fma}\left(b, b \cdot 0.5, -2\right)} \]

   if 2.00000000000000007e154 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a \cdot a, 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot b, \left(b \cdot b\right) \cdot 0.25, -4\right)} \cdot \mathsf{fma}\left(b, b \cdot 0.5, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95) (fma a 0.25 0.5) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95) {
		tmp = fma(a, 0.25, 0.5);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95)
		tmp = fma(a, 0.25, 0.5);
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95], N[(a * 0.25 + 0.5), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.94999999999999996

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 75.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 56.9

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

   8. Applied rewrites 56.9%

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

   if 1.94999999999999996 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 57.3

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 57.3

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

   11. Applied rewrites 57.3%

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

Alternative 14: 39.9% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, 0.25, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma a 0.25 0.5))

C

double code(double a, double b) {
	return fma(a, 0.25, 0.5);
}

Julia

function code(a, b)
	return fma(a, 0.25, 0.5)
end

Wolfram

code[a_, b_] := N[(a * 0.25 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 63.6%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

   3. lower-fma.f64 42.3

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

8. Applied rewrites 42.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]
9. 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 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

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

8. Applied rewrites 41.6%

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