Quotient of sum of exps

Percentage Accurate: 98.9% → 99.3%

Time: 6.8s

Alternatives: 20

Speedup: 2.2×

• could not determine a ground truth

  1. a = 8.724553103329932e+61
  2. b = 3.5361157278208164e+120

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{b} + t\_0}{t\_0}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= (/ (exp a) (+ (exp a) (exp b))) 4e-48)
     (/ (exp a) (+ 1.0 (exp b)))
     (pow (/ (+ (exp b) t_0) t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 4e-48) {
		tmp = exp(a) / (1.0 + exp(b));
	} else {
		tmp = pow(((exp(b) + t_0) / t_0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 4e-48)
		tmp = Float64(exp(a) / Float64(1.0 + exp(b)));
	else
		tmp = Float64(Float64(exp(b) + t_0) / t_0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-48], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Exp[b], $MachinePrecision] + t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 3.9999999999999999e-48

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 99.3%

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

   if 3.9999999999999999e-48 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.2

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

      5. lift-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 99.2

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

   10. Applied rewrites 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{b} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.499999999995:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.499999999995)
   (pow (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b) -1.0)
   (pow (- 2.0 a) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.499999999995) {
		tmp = pow((fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b), -1.0);
	} else {
		tmp = pow((2.0 - a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.499999999995)
		tmp = Float64(fma(Float64(Float64(2.0 / Float64(b * b)) + 0.5), b, 1.0) * b) ^ -1.0;
	else
		tmp = Float64(2.0 - a) ^ -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], 0.499999999995], N[Power[N[(N[(N[(N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999999995

   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 59.9

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

   5. Applied rewrites 59.9%

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

   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 36.5%

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

   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 34.3%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 59.2%

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

   if 0.499999999995 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.4

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

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.3

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 67.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.499999999995:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right), 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.6)
   (pow
    (fma (- 1.0 a) (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) 1.0)
    -1.0)
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.6) {
		tmp = pow(fma((1.0 - a), fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0), 1.0), -1.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.6)
		tmp = fma(Float64(1.0 - a), fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0), 1.0) ^ -1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[Power[N[(N[(1.0 - a), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.599999999999999978

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 76.0

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

   10. Applied rewrites 76.0%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 67.1%

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

   if 0.599999999999999978 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right), 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 99.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. 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 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + e^{b}}{e^{a}}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (pow (/ (+ (fma (fma 0.5 a 1.0) a 1.0) (exp b)) (exp a)) -1.0))

C

double code(double a, double b) {
	return pow(((fma(fma(0.5, a, 1.0), a, 1.0) + exp(b)) / exp(a)), -1.0);
}

Julia

function code(a, b)
	return Float64(Float64(fma(fma(0.5, a, 1.0), a, 1.0) + exp(b)) / exp(a)) ^ -1.0
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.8

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

5. Applied rewrites 97.8%

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 97.8

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

7. Applied rewrites 97.8%

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

8. Final simplification 97.8%

   \[\leadsto {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + e^{b}}{e^{a}}\right)}^{-1} \]
9. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 7: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;a \leq -200000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{b} + t\_0}{t\_0}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= a -200000000.0)
     (/ (exp a) (+ 1.0 (+ 1.0 b)))
     (pow (/ (+ (exp b) t_0) t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if (a <= -200000000.0) {
		tmp = exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = pow(((exp(b) + t_0) / t_0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (a <= -200000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + Float64(1.0 + b)));
	else
		tmp = Float64(Float64(exp(b) + t_0) / t_0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -200000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Exp[b], $MachinePrecision] + t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -2e8 < a

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   8. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 98.9

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

   10. Applied rewrites 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{b} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + e^{b}} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (exp a) (+ (fma (fma 0.5 a 1.0) a 1.0) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (fma(fma(0.5, a, 1.0), a, 1.0) + exp(b));
}

Julia

function code(a, b)
	return Float64(exp(a) / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + exp(b)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.8

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

5. Applied rewrites 97.8%

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
6. Add Preprocessing

Alternative 9: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -180000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, e^{b}, 1\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -180000000.0)
   (/ (exp a) (+ 1.0 (+ 1.0 b)))
   (pow (fma (- 1.0 a) (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -180000000.0) {
		tmp = exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = pow(fma((1.0 - a), exp(b), 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -180000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + Float64(1.0 + b)));
	else
		tmp = fma(Float64(1.0 - a), exp(b), 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -180000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 - a), $MachinePrecision] * N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1.8e8 < a

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -180000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, e^{b}, 1\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -180000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -180000000.0)
   (/ (exp a) (+ 1.0 (+ 1.0 b)))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -180000000.0) {
		tmp = exp(a) / (1.0 + (1.0 + b));
	} 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 (a <= (-180000000.0d0)) then
        tmp = exp(a) / (1.0d0 + (1.0d0 + b))
    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 (a <= -180000000.0) {
		tmp = Math.exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -180000000.0:
		tmp = math.exp(a) / (1.0 + (1.0 + b))
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -180000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + Float64(1.0 + b)));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -180000000.0)
		tmp = exp(a) / (1.0 + (1.0 + b));
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -180000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(1.0 + b), $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 a < -1.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1.8e8 < a

   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 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -180000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -660000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -660000000.0)
   (pow (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b) -1.0)
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -660000000.0) {
		tmp = pow((fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b), -1.0);
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -660000000.0)
		tmp = Float64(fma(Float64(Float64(2.0 / Float64(b * b)) + 0.5), b, 1.0) * b) ^ -1.0;
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -660000000.0], N[Power[N[(N[(N[(N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6e8

   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 35.0

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

   5. Applied rewrites 35.0%

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

   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 22.4%

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

   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 21.6%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 60.7%

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

   if -6.6e8 < a

   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 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -660000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;a \leq -200000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_0 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= a -200000000.0)
     (/ (exp a) (+ 1.0 (+ 1.0 b)))
     (/ t_0 (+ t_0 (exp b))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if (a <= -200000000.0) {
		tmp = exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = t_0 / (t_0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (a <= -200000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + Float64(1.0 + b)));
	else
		tmp = Float64(t_0 / Float64(t_0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -200000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -2e8 < a

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   8. Applied rewrites 98.9%

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

Alternative 13: 58.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right), 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -3.1)
   0.5
   (if (<= b 5.2e+102)
     (pow (fma (- 1.0 a) (fma (fma 0.5 b 1.0) b 1.0) 1.0) -1.0)
     (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -3.1) {
		tmp = 0.5;
	} else if (b <= 5.2e+102) {
		tmp = pow(fma((1.0 - a), fma(fma(0.5, b, 1.0), b, 1.0), 1.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -3.1)
		tmp = 0.5;
	elseif (b <= 5.2e+102)
		tmp = fma(Float64(1.0 - a), fma(fma(0.5, b, 1.0), b, 1.0), 1.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3.1], 0.5, If[LessEqual[b, 5.2e+102], N[Power[N[(N[(1.0 - a), $MachinePrecision] * N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.10000000000000009

   1. Initial program 97.8%

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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -3.10000000000000009 < b < 5.20000000000000013e102

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 68.0

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

   10. Applied rewrites 68.0%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 55.6%

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

   if 5.20000000000000013e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right), 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -125000000:\\ \;\;\;\;\frac{e^{a}}{1 + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(e^{b} + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -125000000.0)
   (/ (exp a) (+ 1.0 (+ 1.0 b)))
   (/ (+ 1.0 a) (+ (+ (exp b) a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -125000000.0) {
		tmp = exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = (1.0 + a) / ((exp(b) + a) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-125000000.0d0)) then
        tmp = exp(a) / (1.0d0 + (1.0d0 + b))
    else
        tmp = (1.0d0 + a) / ((exp(b) + a) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -125000000.0) {
		tmp = Math.exp(a) / (1.0 + (1.0 + b));
	} else {
		tmp = (1.0 + a) / ((Math.exp(b) + a) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -125000000.0:
		tmp = math.exp(a) / (1.0 + (1.0 + b))
	else:
		tmp = (1.0 + a) / ((math.exp(b) + a) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -125000000.0)
		tmp = Float64(exp(a) / Float64(1.0 + Float64(1.0 + b)));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(exp(b) + a) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -125000000.0)
		tmp = exp(a) / (1.0 + (1.0 + b));
	else
		tmp = (1.0 + a) / ((exp(b) + a) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -125000000.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[Exp[b], $MachinePrecision] + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1.25e8 < a

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 94.0

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

   8. Applied rewrites 94.0%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. lower-exp.f64 98.4

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

   11. Applied rewrites 98.4%

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

Alternative 15: 54.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+118}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, 1 + b, 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.25)
   0.5
   (if (<= b 2e+118)
     (pow (fma (- 1.0 a) (+ 1.0 b) 1.0) -1.0)
     (pow (* (* b b) 0.5) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 0.5;
	} else if (b <= 2e+118) {
		tmp = pow(fma((1.0 - a), (1.0 + b), 1.0), -1.0);
	} else {
		tmp = pow(((b * b) * 0.5), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.25)
		tmp = 0.5;
	elseif (b <= 2e+118)
		tmp = fma(Float64(1.0 - a), Float64(1.0 + b), 1.0) ^ -1.0;
	else
		tmp = Float64(Float64(b * b) * 0.5) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.25], 0.5, If[LessEqual[b, 2e+118], N[Power[N[(N[(1.0 - a), $MachinePrecision] * N[(1.0 + b), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25

   1. Initial program 97.8%

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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1.25 < b < 1.99999999999999993e118

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 68.8

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

   10. Applied rewrites 68.8%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 53.9%

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

   if 1.99999999999999993e118 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+118}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1 - a, 1 + b, 1\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-15)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e-15) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e-15)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e-15], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e-15

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 71.4

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

   10. Applied rewrites 71.4%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 49.9%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 1.5e-15 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-15)
   (pow (- 2.0 a) -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 <= 1.5e-15) {
		tmp = pow((2.0 - a), -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 <= 1.5e-15)
		tmp = Float64(2.0 - a) ^ -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, 1.5e-15], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], 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 < 1.5e-15

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 71.4

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

   10. Applied rewrites 71.4%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 49.9%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 1.5e-15 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+54) (pow (- 2.0 a) -1.0) (pow (* (* b b) 0.5) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+54) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((b * b) * 0.5), -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 (b <= 1.15d+54) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((b * b) * 0.5d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+54) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((b * b) * 0.5), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.15e+54:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((b * b) * 0.5), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e+54)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(b * b) * 0.5) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e+54)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((b * b) * 0.5) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e+54], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999997e54

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 46.7%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 1.14999999999999997e54 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 72.4%

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

   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 72.4%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 - a\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return math.pow((2.0 - a), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.8

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

5. Applied rewrites 97.8%

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + e^{b}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 97.8

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-rgt1-in N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 79.7

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

10. Applied rewrites 79.7%

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

11. Taylor expanded in b around 0

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

13. Applied rewrites 37.0%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

14. Final simplification 37.0%

    \[\leadsto {\left(2 - a\right)}^{-1} \]
15. Add Preprocessing

Alternative 20: 39.2% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-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 78.7

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

5. Applied rewrites 78.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 35.9%

   \[\leadsto 0.5 \]
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 2024312 
(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))))