Quotient of sum of exps

Percentage Accurate: 99.0% → 99.1%

Time: 7.0s

Alternatives: 20

Speedup: 1.0×

• could not determine a ground truth

  1. a = 4.510830927822325e+143
  2. b = 2.974566907786603e-118

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. inv-pow N/A

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

   5. pow-to-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-log.f64 99.2

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{elif}\;t\_0 \leq 0.5001999411546286:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 0.0)
     (* (* 0.020833333333333332 (* b b)) b)
     (if (<= t_0 0.5001999411546286)
       (fma (fma (* b b) 0.020833333333333332 -0.25) b 0.5)
       1.0))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else if (t_0 <= 0.5001999411546286) {
		tmp = fma(fma((b * b), 0.020833333333333332, -0.25), b, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	elseif (t_0 <= 0.5001999411546286)
		tmp = fma(fma(Float64(b * b), 0.020833333333333332, -0.25), b, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t$95$0, 0.5001999411546286], N[(N[(N[(b * b), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * b + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 27.4%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 27.4%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500199941154628624

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.7%

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

   if 0.500199941154628624 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 96.0%

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

Alternative 3: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t\_0 \leq 10^{-27}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{elif}\;t\_0 \leq 0.5001999411546286:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 1e-27)
     (* (* 0.020833333333333332 (* b b)) b)
     (if (<= t_0 0.5001999411546286) (fma -0.25 b 0.5) 1.0))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 1e-27) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else if (t_0 <= 0.5001999411546286) {
		tmp = fma(-0.25, b, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 1e-27)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	elseif (t_0 <= 0.5001999411546286)
		tmp = fma(-0.25, b, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-27], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t$95$0, 0.5001999411546286], N[(-0.25 * b + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 1e-27

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 27.2%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 27.2%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if 1e-27 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500199941154628624

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.1%

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

   if 0.500199941154628624 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 96.0%

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

Alternative 4: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.5001999411546286:\\ \;\;\;\;{\left(2 + b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.5001999411546286)
   (pow (+ 2.0 b) -1.0)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286) {
		tmp = pow((2.0 + b), -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286d0) then
        tmp = (2.0d0 + b) ** (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(a) + Math.exp(b))) <= 0.5001999411546286) {
		tmp = Math.pow((2.0 + b), -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(a) + math.exp(b))) <= 0.5001999411546286:
		tmp = math.pow((2.0 + b), -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.5001999411546286)
		tmp = Float64(2.0 + b) ^ -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286)
		tmp = (2.0 + b) ^ -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5001999411546286], N[Power[N[(2.0 + b), $MachinePrecision], -1.0], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500199941154628624

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.1%

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

   if 0.500199941154628624 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 96.0%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.5001999411546286:\\ \;\;\;\;{\left(2 + b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999)
   (exp (- a (log1p (exp a))))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999) {
		tmp = exp((a - log1p(exp(a))));
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.9999) {
		tmp = Math.exp((a - Math.log1p(Math.exp(a))));
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9999:
		tmp = math.exp((a - math.log1p(math.exp(a))))
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999)
		tmp = exp(Float64(a - log1p(exp(a))));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999], N[Exp[N[(a - N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 98.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-log1p.f64 N/A

         \[\leadsto e^{a - \color{blue}{\mathsf{log1p}\left(e^{a}\right)}} \]

      5. lower-exp.f64 98.7

         \[\leadsto e^{a - \mathsf{log1p}\left(\color{blue}{e^{a}}\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto e^{\color{blue}{a - \mathsf{log1p}\left(e^{a}\right)}} \]

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;e^{a - \mathsf{log1p}\left(e^{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.5001999411546286:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.5001999411546286)
   (fma -0.25 b 0.5)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286) {
		tmp = fma(-0.25, b, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.5001999411546286)
		tmp = fma(-0.25, b, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5001999411546286], N[(-0.25 * b + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500199941154628624

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 41.0%

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

   if 0.500199941154628624 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 96.0%

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

Alternative 7: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.0)
   1.0
   (if (<= (exp b) 2.0)
     (fma (fma (* b b) 0.020833333333333332 -0.25) b 0.5)
     (pow (* (* 0.5 b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.0) {
		tmp = 1.0;
	} else if (exp(b) <= 2.0) {
		tmp = fma(fma((b * b), 0.020833333333333332, -0.25), b, 0.5);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.0)
		tmp = 1.0;
	elseif (exp(b) <= 2.0)
		tmp = fma(fma(Float64(b * b), 0.020833333333333332, -0.25), b, 0.5);
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.0], 1.0, If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], N[(N[(N[(b * b), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * b + 0.5), $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.0

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 97.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (exp.f64 b) < 2

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

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

   if 2 < (exp.f64 b)

   1. Initial program 98.7%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 55.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.9999d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.5001999411546286:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.5001999411546286) 0.5 1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(a) + Math.exp(b))) <= 0.5001999411546286) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(a) + math.exp(b))) <= 0.5001999411546286:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.5001999411546286)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.5001999411546286)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5001999411546286], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500199941154628624

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 40.8%

      \[\leadsto 0.5 \]

   if 0.500199941154628624 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 96.0%

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

Alternative 10: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 11: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999)
   (/ (exp a) (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999) {
		tmp = exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0);
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999)
		tmp = Float64(exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999], N[(N[Exp[a], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.7%

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

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{a}}{2} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0)
   1.0
   (if (<= b 9.5e+102)
     (/ (exp a) 2.0)
     (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else if (b <= 9.5e+102) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	elseif (b <= 9.5e+102)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, If[LessEqual[b, 9.5e+102], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b < 9.4999999999999992e102

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.0%

      \[\leadsto \frac{e^{a}}{2} \]

   if 9.4999999999999992e102 < b

   1. Initial program 98.0%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+75}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0)
   1.0
   (if (<= b 1.35e+75)
     (/ (+ 1.0 a) (fma (fma 0.5 a 1.0) a 2.0))
     (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else if (b <= 1.35e+75) {
		tmp = (1.0 + a) / fma(fma(0.5, a, 1.0), a, 2.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	elseif (b <= 1.35e+75)
		tmp = Float64(Float64(1.0 + a) / fma(fma(0.5, a, 1.0), a, 2.0));
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, If[LessEqual[b, 1.35e+75], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b < 1.34999999999999999e75

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 75.1

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

   11. Applied rewrites 75.1%

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

   if 1.34999999999999999e75 < b

   1. Initial program 98.2%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 90.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+75}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.2)
   1.0
   (if (<= b 2.4)
     (fma (fma (* b b) 0.020833333333333332 -0.25) b 0.5)
     (pow (* (fma 0.5 b 1.0) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.2) {
		tmp = 1.0;
	} else if (b <= 2.4) {
		tmp = fma(fma((b * b), 0.020833333333333332, -0.25), b, 0.5);
	} else {
		tmp = pow((fma(0.5, b, 1.0) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.2)
		tmp = 1.0;
	elseif (b <= 2.4)
		tmp = fma(fma(Float64(b * b), 0.020833333333333332, -0.25), b, 0.5);
	else
		tmp = Float64(fma(0.5, b, 1.0) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.2], 1.0, If[LessEqual[b, 2.4], N[(N[(N[(b * b), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * b + 0.5), $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2000000000000002

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 97.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 97.9%

      \[\leadsto \color{blue}{1} \]

   if -2.2000000000000002 < b < 2.39999999999999991

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

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

   if 2.39999999999999991 < b

   1. Initial program 98.7%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 55.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0)
   1.0
   (if (<= b 2.4e+140)
     (/ (+ 1.0 a) (fma (fma 0.5 a 1.0) a 2.0))
     (pow (* (* 0.5 b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else if (b <= 2.4e+140) {
		tmp = (1.0 + a) / fma(fma(0.5, a, 1.0), a, 2.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	elseif (b <= 2.4e+140)
		tmp = Float64(Float64(1.0 + a) / fma(fma(0.5, a, 1.0), a, 2.0));
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, If[LessEqual[b, 2.4e+140], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b < 2.4e140

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 70.6

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

   11. Applied rewrites 70.6%

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

   if 2.4e140 < b

   1. Initial program 97.7%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 93.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0)
   1.0
   (if (<= b 2.4e+140)
     (/ 1.0 (fma (fma 0.5 a 1.0) a 2.0))
     (pow (* (* 0.5 b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else if (b <= 2.4e+140) {
		tmp = 1.0 / fma(fma(0.5, a, 1.0), a, 2.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	elseif (b <= 2.4e+140)
		tmp = Float64(1.0 / fma(fma(0.5, a, 1.0), a, 2.0));
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, If[LessEqual[b, 2.4e+140], N[(1.0 / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b < 2.4e140

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.0%

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

   if 2.4e140 < b

   1. Initial program 97.7%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 93.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0) 1.0 (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b

   1. Initial program 98.6%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7800.0) 1.0 (pow (fma (* 0.5 b) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7800.0) {
		tmp = 1.0;
	} else {
		tmp = pow(fma((0.5 * b), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7800.0)
		tmp = 1.0;
	else
		tmp = fma(Float64(0.5 * b), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7800.0], 1.0, N[Power[N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7800

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. inv-pow N/A

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

      5. pow-to-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \]

   if -7800 < b

   1. Initial program 98.6%

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

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 59.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.1% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 80.4

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.5%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024321 
(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))))