Quotient of sum of exps

Percentage Accurate: 98.9% → 99.3%

Time: 9.0s

Alternatives: 17

Speedup: 1.2×

• could not determine a ground truth

  1. a = 1.885141171662937e+195
  2. b = 5.050207789237553e-153

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, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t\_0, 1\right)}{\mathsf{fma}\left(a, t\_0, e^{b} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a (fma a 0.16666666666666666 0.5) 1.0)))
   (if (<= (exp a) 0.0)
     (/ (exp a) (+ 1.0 1.0))
     (/ (fma a t_0 1.0) (fma a t_0 (+ (exp b) 1.0))))))

C

double code(double a, double b) {
	double t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = fma(a, t_0, 1.0) / fma(a, t_0, (exp(b) + 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(fma(a, t_0, 1.0) / fma(a, t_0, Float64(exp(b) + 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * t$95$0 + 1.0), $MachinePrecision] / N[(a * t$95$0 + N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), e^{b} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(\frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}} - -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t\_0, 1\right)}{\mathsf{fma}\left(a, t\_0, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a (fma a 0.16666666666666666 0.5) 1.0)))
   (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
     (/
      1.0
      (*
       (* b b)
       (*
        b
        (-
         (/
          (* (+ 0.125 (/ 1.0 (* b (* b b)))) (/ 1.0 b))
          (+ 0.25 (/ (- (/ 1.0 b) 0.5) b)))
         -0.16666666666666666))))
     (/ (fma a t_0 1.0) (fma a t_0 2.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / ((b * b) * (b * ((((0.125 + (1.0 / (b * (b * b)))) * (1.0 / b)) / (0.25 + (((1.0 / b) - 0.5) / b))) - -0.16666666666666666)));
	} else {
		tmp = fma(a, t_0, 1.0) / fma(a, t_0, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(b * b) * Float64(b * Float64(Float64(Float64(Float64(0.125 + Float64(1.0 / Float64(b * Float64(b * b)))) * Float64(1.0 / b)) / Float64(0.25 + Float64(Float64(Float64(1.0 / b) - 0.5) / b))) - -0.16666666666666666))));
	else
		tmp = Float64(fma(a, t_0, 1.0) / fma(a, t_0, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(b * N[(N[(N[(N[(0.125 + N[(1.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(N[(1.0 / b), $MachinePrecision] - 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * t$95$0 + 1.0), $MachinePrecision] / N[(a * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 45.2%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 47.3%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(-\left(-0.16666666666666666 - \frac{0.5 + \frac{1}{b}}{b}\right)\right)}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(-\left(-0.16666666666666666 - \frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}}\right)\right)\right)} \]

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

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 67.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(\frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}} - -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(b, 0.5, 1\right) + \left(b \cdot b\right) \cdot \left(0.16666666666666666 + \frac{2}{b \cdot \left(b \cdot b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t\_0, 1\right)}{\mathsf{fma}\left(a, t\_0, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a (fma a 0.16666666666666666 0.5) 1.0)))
   (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
     (/
      1.0
      (*
       b
       (+
        (fma b 0.5 1.0)
        (* (* b b) (+ 0.16666666666666666 (/ 2.0 (* b (* b b))))))))
     (/ (fma a t_0 1.0) (fma a t_0 2.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / (b * (fma(b, 0.5, 1.0) + ((b * b) * (0.16666666666666666 + (2.0 / (b * (b * b)))))));
	} else {
		tmp = fma(a, t_0, 1.0) / fma(a, t_0, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, fma(a, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / Float64(b * Float64(fma(b, 0.5, 1.0) + Float64(Float64(b * b) * Float64(0.16666666666666666 + Float64(2.0 / Float64(b * Float64(b * b))))))));
	else
		tmp = Float64(fma(a, t_0, 1.0) / fma(a, t_0, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(b * N[(N[(b * 0.5 + 1.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.16666666666666666 + N[(2.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * t$95$0 + 1.0), $MachinePrecision] / N[(a * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 45.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 52.9%

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

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

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 67.5%

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

Alternative 4: 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 5: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

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

Alternative 6: 57.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.5

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 53.6%

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

   if 1 < (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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.7%

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

4. Final simplification 58.6%

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

Alternative 7: 57.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.5

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 53.6%

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

   if 1 < (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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 73.7%

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

4. Final simplification 58.6%

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

Alternative 8: 53.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.5

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 53.6%

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

   if 1 < (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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.2%

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

4. Final simplification 54.5%

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

Alternative 9: 81.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(\frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}} - -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+133)
   (/
    1.0
    (*
     (* b b)
     (*
      b
      (-
       (/
        (* (+ 0.125 (/ 1.0 (* b (* b b)))) (/ 1.0 b))
        (+ 0.25 (/ (- (/ 1.0 b) 0.5) b)))
       -0.16666666666666666))))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+133) {
		tmp = 1.0 / ((b * b) * (b * ((((0.125 + (1.0 / (b * (b * b)))) * (1.0 / b)) / (0.25 + (((1.0 / b) - 0.5) / b))) - -0.16666666666666666)));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5d+133)) then
        tmp = 1.0d0 / ((b * b) * (b * ((((0.125d0 + (1.0d0 / (b * (b * b)))) * (1.0d0 / b)) / (0.25d0 + (((1.0d0 / b) - 0.5d0) / b))) - (-0.16666666666666666d0))))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+133) {
		tmp = 1.0 / ((b * b) * (b * ((((0.125 + (1.0 / (b * (b * b)))) * (1.0 / b)) / (0.25 + (((1.0 / b) - 0.5) / b))) - -0.16666666666666666)));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+133:
		tmp = 1.0 / ((b * b) * (b * ((((0.125 + (1.0 / (b * (b * b)))) * (1.0 / b)) / (0.25 + (((1.0 / b) - 0.5) / b))) - -0.16666666666666666)))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+133)
		tmp = Float64(1.0 / Float64(Float64(b * b) * Float64(b * Float64(Float64(Float64(Float64(0.125 + Float64(1.0 / Float64(b * Float64(b * b)))) * Float64(1.0 / b)) / Float64(0.25 + Float64(Float64(Float64(1.0 / b) - 0.5) / b))) - -0.16666666666666666))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+133)
		tmp = 1.0 / ((b * b) * (b * ((((0.125 + (1.0 / (b * (b * b)))) * (1.0 / b)) / (0.25 + (((1.0 / b) - 0.5) / b))) - -0.16666666666666666)));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+133], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(b * N[(N[(N[(N[(0.125 + N[(1.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(N[(1.0 / b), $MachinePrecision] - 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999961e133

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 29.6%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 33.5%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(-\left(-0.16666666666666666 - \frac{0.5 + \frac{1}{b}}{b}\right)\right)}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 52.4%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(-\left(-0.16666666666666666 - \frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}}\right)\right)\right)} \]

   if -4.99999999999999961e133 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(\frac{\left(0.125 + \frac{1}{b \cdot \left(b \cdot b\right)}\right) \cdot \frac{1}{b}}{0.25 + \frac{\frac{1}{b} - 0.5}{b}} - -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, t\_0 \cdot t\_0, -4\right)}{\mathsf{fma}\left(b, t\_0, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma 0.16666666666666666 b 0.5) 1.0)))
   (if (<= b -6.6e-206)
     (/
      (fma a (fma a 0.5 1.0) 1.0)
      (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
     (if (<= b 5e+102)
       (/ 1.0 (/ (fma (* b b) (* t_0 t_0) -4.0) (fma b t_0 -2.0)))
       (/ 1.0 (* 0.16666666666666666 (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = fma(b, fma(0.16666666666666666, b, 0.5), 1.0);
	double tmp;
	if (b <= -6.6e-206) {
		tmp = fma(a, fma(a, 0.5, 1.0), 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 5e+102) {
		tmp = 1.0 / (fma((b * b), (t_0 * t_0), -4.0) / fma(b, t_0, -2.0));
	} else {
		tmp = 1.0 / (0.16666666666666666 * (b * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, fma(0.16666666666666666, b, 0.5), 1.0)
	tmp = 0.0
	if (b <= -6.6e-206)
		tmp = Float64(fma(a, fma(a, 0.5, 1.0), 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 5e+102)
		tmp = Float64(1.0 / Float64(fma(Float64(b * b), Float64(t_0 * t_0), -4.0) / fma(b, t_0, -2.0)));
	else
		tmp = Float64(1.0 / Float64(0.16666666666666666 * Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(0.16666666666666666 * b + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, -6.6e-206], N[(N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+102], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -4.0), $MachinePrecision] / N[(b * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.16666666666666666 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.59999999999999961e-206

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 51.9

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 39.9%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if -6.59999999999999961e-206 < b < 5e102

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.7%

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

   10. Applied rewrites 66.6%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right), -4\right)}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right)}, -2\right)}} \]

   if 5e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{1}{0.16666666666666666 \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right), -4\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b \cdot b, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.6e-206)
   (/
    (fma a (fma a 0.5 1.0) 1.0)
    (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= b 5e+153)
     (/
      1.0
      (fma
       b
       (/
        (fma
         (fma 0.16666666666666666 b 0.5)
         (* (* b b) (fma 0.16666666666666666 b 0.5))
         -1.0)
        (fma b (fma 0.16666666666666666 b 0.5) -1.0))
       2.0))
     (/ 1.0 (fma 0.5 (* b b) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.6e-206) {
		tmp = fma(a, fma(a, 0.5, 1.0), 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 5e+153) {
		tmp = 1.0 / fma(b, (fma(fma(0.16666666666666666, b, 0.5), ((b * b) * fma(0.16666666666666666, b, 0.5)), -1.0) / fma(b, fma(0.16666666666666666, b, 0.5), -1.0)), 2.0);
	} else {
		tmp = 1.0 / fma(0.5, (b * b), b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.6e-206)
		tmp = Float64(fma(a, fma(a, 0.5, 1.0), 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 5e+153)
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(0.16666666666666666, b, 0.5), Float64(Float64(b * b) * fma(0.16666666666666666, b, 0.5)), -1.0) / fma(b, fma(0.16666666666666666, b, 0.5), -1.0)), 2.0));
	else
		tmp = Float64(1.0 / fma(0.5, Float64(b * b), b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.6e-206], N[(N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+153], N[(1.0 / N[(b * N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(b * N[(0.16666666666666666 * b + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.59999999999999961e-206

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 51.9

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 39.9%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if -6.59999999999999961e-206 < b < 5.00000000000000018e153

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 61.4%

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

   10. Applied rewrites 68.8%

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

   if 5.00000000000000018e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(-\left(-0.16666666666666666 - \frac{0.5 + \frac{1}{b}}{b}\right)\right)}\right)} \]

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 100.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b \cdot b, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.6e-206)
   (/
    (fma a (fma a 0.5 1.0) 1.0)
    (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.6e-206) {
		tmp = fma(a, fma(a, 0.5, 1.0), 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.6e-206)
		tmp = Float64(fma(a, fma(a, 0.5, 1.0), 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.6e-206], N[(N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.59999999999999961e-206

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 51.9

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 39.9%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if -6.59999999999999961e-206 < b

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.7%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.1e+49)
   (/ 1.0 (+ 1.0 (+ a 1.0)))
   (/ 1.0 (* (* b b) (fma b 0.16666666666666666 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.1e+49) {
		tmp = 1.0 / (1.0 + (a + 1.0));
	} else {
		tmp = 1.0 / ((b * b) * fma(b, 0.16666666666666666, 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.1e+49)
		tmp = Float64(1.0 / Float64(1.0 + Float64(a + 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * fma(b, 0.16666666666666666, 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.1e+49], N[(1.0 / N[(1.0 + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1e49

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 51.9%

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

   if 4.1e49 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 82.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.1e+49)
   (/ 1.0 (+ 1.0 (+ a 1.0)))
   (/ 1.0 (* 0.16666666666666666 (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.1e+49) {
		tmp = 1.0 / (1.0 + (a + 1.0));
	} else {
		tmp = 1.0 / (0.16666666666666666 * (b * (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 4.1d+49) then
        tmp = 1.0d0 / (1.0d0 + (a + 1.0d0))
    else
        tmp = 1.0d0 / (0.16666666666666666d0 * (b * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.1e+49) {
		tmp = 1.0 / (1.0 + (a + 1.0));
	} else {
		tmp = 1.0 / (0.16666666666666666 * (b * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.1e+49:
		tmp = 1.0 / (1.0 + (a + 1.0))
	else:
		tmp = 1.0 / (0.16666666666666666 * (b * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.1e+49)
		tmp = Float64(1.0 / Float64(1.0 + Float64(a + 1.0)));
	else
		tmp = Float64(1.0 / Float64(0.16666666666666666 * Float64(b * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.1e+49)
		tmp = 1.0 / (1.0 + (a + 1.0));
	else
		tmp = 1.0 / (0.16666666666666666 * (b * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.1e+49], N[(1.0 / N[(1.0 + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.16666666666666666 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1e49

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 51.9%

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

   if 4.1e49 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.0%

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

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.2% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b \cdot b, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.1e+49) (/ 1.0 (+ 1.0 (+ a 1.0))) (/ 1.0 (fma 0.5 (* b b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.1e+49) {
		tmp = 1.0 / (1.0 + (a + 1.0));
	} else {
		tmp = 1.0 / fma(0.5, (b * b), b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.1e+49)
		tmp = Float64(1.0 / Float64(1.0 + Float64(a + 1.0)));
	else
		tmp = Float64(1.0 / fma(0.5, Float64(b * b), b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.1e+49], N[(1.0 / N[(1.0 + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1e49

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 51.9%

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

   if 4.1e49 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 82.0%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(-\left(-0.16666666666666666 - \frac{0.5 + \frac{1}{b}}{b}\right)\right)}\right)} \]

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 63.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{1 + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b \cdot b, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(a + 1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / (1.0 + (a + 1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in a around 0

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

   1. lower-+.f64 66.3

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

8. Applied rewrites 66.3%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 41.5%

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

12. Final simplification 41.5%

    \[\leadsto \frac{1}{1 + \left(a + 1\right)} \]
13. Add Preprocessing

Alternative 17: 39.4% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 82.8

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 40.8%

   \[\leadsto 0.5 \]
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 2024238 
(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))))