Quotient of sum of exps

Percentage Accurate: 98.9% → 99.3%

Time: 8.2s

Alternatives: 17

Speedup: 1.3×

• could not determine a ground truth

  1. a = 1.6607956168381566e+40
  2. b = 5.769358818889093e+58

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(a * N[(a * N[(0.16666666666666666 * a + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 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 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 100.0

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-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. +-lowering-+.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. exp-lowering-exp.f64 97.5

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

   5. Simplified 97.5%

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

   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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. rgt-mult-inverse N/A

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

      11. distribute-lft-in N/A

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

      12. associate-+r+ N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 99.4%

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

4. Final simplification 99.5%

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

Alternative 2: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.7075496127730688:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.7075496127730688)
   (fma 0.25 a 0.5)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.7075496127730688) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.7075496127730688)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.7075496127730688], N[(0.25 * a + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.707549612773068826

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 74.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 50.3

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

   8. Simplified 50.3%

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

   if 0.707549612773068826 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.6

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

   5. Simplified 92.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 97.9%

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

Alternative 3: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.7075496127730688:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.7075496127730688)
   (fma -0.25 b 0.5)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.7075496127730688) {
		tmp = fma(-0.25, b, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.7075496127730688)
		tmp = fma(-0.25, b, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.7075496127730688], N[(-0.25 * b + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.707549612773068826

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 50.3

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

   8. Simplified 50.3%

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

   if 0.707549612773068826 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.6

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

   5. Simplified 92.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 97.9%

       \[\leadsto \color{blue}{1} \]
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 98.8%

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

Alternative 5: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(0.5 * a + 1.0), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 100.0

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 99.3

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

   8. Simplified 99.3%

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

Alternative 6: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) * 0.5;
	} 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) * 0.5d0
    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) * 0.5;
	} 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) * 0.5
	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) * 0.5);
	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) * 0.5;
	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] * 0.5), $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 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 100.0

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 98.4

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

   5. Simplified 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.999:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.999)
   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) <= 0.999) {
		tmp = 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) <= 0.999)
		tmp = 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], 0.999], 1.0, 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) < 0.998999999999999999

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if 0.998999999999999999 < (exp.f64 b)

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 72.2

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

   8. Simplified 72.2%

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

Alternative 8: 93.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{-0.5}, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 2.3e+67)
     (* (exp a) 0.5)
     (/
      1.0
      (fma
       b
       (fma b (/ (fma (* b b) 0.027777777777777776 -0.25) -0.5) 1.0)
       2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 2.3e+67) {
		tmp = exp(a) * 0.5;
	} else {
		tmp = 1.0 / fma(b, fma(b, (fma((b * b), 0.027777777777777776, -0.25) / -0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 2.3e+67)
		tmp = Float64(exp(a) * 0.5);
	else
		tmp = Float64(1.0 / fma(b, fma(b, Float64(fma(Float64(b * b), 0.027777777777777776, -0.25) / -0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 2.3e+67], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(N[(N[(b * b), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision] / -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 2.2999999999999999e67

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 93.2%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 93.2

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 93.2%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 2.2999999999999999e67 < b

   1. Initial program 98.1%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 88.1

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

   8. Simplified 88.1%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 88.1

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(b, 0.16666666666666666, \color{blue}{-0.5}\right)}, 1\right), 2\right)} \]

   10. Applied egg-rr 88.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(b, 0.16666666666666666, -0.5\right)}}, 1\right), 2\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{-1}{4}\right)}{\color{blue}{\frac{-1}{2}}}, 1\right), 2\right)} \]
   12. Step-by-step derivation

   13. Simplified 98.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\color{blue}{-0.5}}, 1\right), 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.999:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (exp b) 0.999) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.999) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 0.999d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 0.999) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(b) <= 0.999:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.999)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 0.999)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.999], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.998999999999999999

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if 0.998999999999999999 < (exp.f64 b)

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 49.5%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{-0.5}, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (/
    1.0
    (fma
     b
     (fma b (/ (fma (* b b) 0.027777777777777776 -0.25) -0.5) 1.0)
     2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / fma(b, fma(b, (fma((b * b), 0.027777777777777776, -0.25) / -0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / fma(b, fma(b, Float64(fma(Float64(b * b), 0.027777777777777776, -0.25) / -0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, N[(1.0 / N[(b * N[(b * N[(N[(N[(b * b), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision] / -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 72.2

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

   8. Simplified 72.2%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 72.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(b, 0.16666666666666666, \color{blue}{-0.5}\right)}, 1\right), 2\right)} \]

   10. Applied egg-rr 72.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\mathsf{fma}\left(b, 0.16666666666666666, -0.5\right)}}, 1\right), 2\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, \frac{1}{36}, \frac{-1}{4}\right)}{\color{blue}{\frac{-1}{2}}}, 1\right), 2\right)} \]
   12. Step-by-step derivation

   13. Simplified 74.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{\mathsf{fma}\left(b \cdot b, 0.027777777777777776, -0.25\right)}{\color{blue}{-0.5}}, 1\right), 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 2.5)
     (fma
      b
      (fma
       (* b b)
       (fma (* b b) -0.0020833333333333333 0.020833333333333332)
       -0.25)
      0.5)
     (if (<= b 5.6e+102)
       (* a (* a (* a -0.020833333333333332)))
       (/ 6.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 2.5) {
		tmp = fma(b, fma((b * b), fma((b * b), -0.0020833333333333333, 0.020833333333333332), -0.25), 0.5);
	} else if (b <= 5.6e+102) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 2.5)
		tmp = fma(b, fma(Float64(b * b), fma(Float64(b * b), -0.0020833333333333333, 0.020833333333333332), -0.25), 0.5);
	elseif (b <= 5.6e+102)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 2.5], N[(b * N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -0.0020833333333333333 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 5.6e+102], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 2.5

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 71.7

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

   8. Simplified 71.7%

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

   if 2.5 < b < 5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 35.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 41.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 41.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 5.60000000000000037e102 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 340:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, b \cdot 0.020833333333333332, -0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 340.0)
     (fma b (fma b (* b 0.020833333333333332) -0.25) 0.5)
     (if (<= b 5.6e+102)
       (* a (* a (* a -0.020833333333333332)))
       (/ 6.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 340.0) {
		tmp = fma(b, fma(b, (b * 0.020833333333333332), -0.25), 0.5);
	} else if (b <= 5.6e+102) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 340.0)
		tmp = fma(b, fma(b, Float64(b * 0.020833333333333332), -0.25), 0.5);
	elseif (b <= 5.6e+102)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 340.0], N[(b * N[(b * N[(b * 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 5.6e+102], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 340

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 71.5

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

   8. Simplified 71.5%

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

   if 340 < b < 5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 35.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 41.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 41.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 5.60000000000000037e102 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 340:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, b \cdot 0.020833333333333332, -0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+149}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 340.0)
     (fma b (fma b (* b 0.020833333333333332) -0.25) 0.5)
     (if (<= b 9e+149)
       (* a (* a (* a -0.020833333333333332)))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 340.0) {
		tmp = fma(b, fma(b, (b * 0.020833333333333332), -0.25), 0.5);
	} else if (b <= 9e+149) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 340.0)
		tmp = fma(b, fma(b, Float64(b * 0.020833333333333332), -0.25), 0.5);
	elseif (b <= 9e+149)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 340.0], N[(b * N[(b * N[(b * 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 9e+149], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 340

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 71.5

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

   8. Simplified 71.5%

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

   if 340 < b < 8.99999999999999965e149

   1. Initial program 97.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 25.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 45.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 45.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 8.99999999999999965e149 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 91.2

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

   8. Simplified 91.2%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 91.2

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

   11. Simplified 91.2%

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

Alternative 14: 72.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+149}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 2.0)
     (fma -0.25 b 0.5)
     (if (<= b 9e+149)
       (* a (* a (* a -0.020833333333333332)))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 2.0) {
		tmp = fma(-0.25, b, 0.5);
	} else if (b <= 9e+149) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 2.0)
		tmp = fma(-0.25, b, 0.5);
	elseif (b <= 9e+149)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 2.0], N[(-0.25 * b + 0.5), $MachinePrecision], If[LessEqual[b, 9e+149], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 2

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 71.3

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

   8. Simplified 71.3%

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

   if 2 < b < 8.99999999999999965e149

   1. Initial program 97.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 25.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 45.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 45.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 8.99999999999999965e149 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 91.2

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

   8. Simplified 91.2%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 91.2

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

   11. Simplified 91.2%

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

Alternative 15: 65.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.0011)
   1.0
   (if (<= b 2.0) (fma -0.25 b 0.5) (* a (* a (* a -0.020833333333333332))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else if (b <= 2.0) {
		tmp = fma(-0.25, b, 0.5);
	} else {
		tmp = a * (a * (a * -0.020833333333333332));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	elseif (b <= 2.0)
		tmp = fma(-0.25, b, 0.5);
	else
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, If[LessEqual[b, 2.0], N[(-0.25 * b + 0.5), $MachinePrecision], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b < 2

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 71.3

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

   8. Simplified 71.3%

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

   if 2 < b

   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. Simplified 21.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.9

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

   8. Simplified 2.9%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 51.5

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 51.5%

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

Alternative 16: 54.5% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0011:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.0011) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.0011d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.0011) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.0011:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.0011)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.0011)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.0011], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00110000000000000007

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 98.7

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 100.0%

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

   if -0.00110000000000000007 < b

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 80.5

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

   5. Simplified 80.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 50.8

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

   8. Simplified 50.8%

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

Alternative 17: 38.7% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. exp-lowering-exp.f64 83.6

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

5. Simplified 83.6%

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

6. Taylor expanded in b around 0

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

8. Simplified 44.1%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(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))))