Quotient of sum of exps

Percentage Accurate: 98.9% → 98.6%

Time: 7.3s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.2548942082943142e+157
  2. b = 9.466879054583993e+46

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: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) * 0.5d0
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) * 0.5;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	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 / (1.0 + math.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(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a) * 0.5;
	else
		tmp = 1.0 / (1.0 + exp(b));
	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[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{1}} \]
   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 98.9%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. clear-num N/A

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

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

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

   3. div-inv N/A

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

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

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

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

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

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

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

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

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

   8. rec-exp N/A

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

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

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

   10. neg-lowering-neg.f64 98.8

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

4. Applied egg-rr 98.8%

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

Alternative 3: 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 4: 78.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+53}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.6e+53)
   (* (exp a) 0.5)
   (if (<= b 1.35e+154)
     (* (/ (+ b -2.0) (fma b (* b (* b b)) -16.0)) (fma b b 4.0))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.6e+53) {
		tmp = exp(a) * 0.5;
	} else if (b <= 1.35e+154) {
		tmp = ((b + -2.0) / fma(b, (b * (b * b)), -16.0)) * fma(b, b, 4.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.6e+53)
		tmp = Float64(exp(a) * 0.5);
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(Float64(b + -2.0) / fma(b, Float64(b * Float64(b * b)), -16.0)) * fma(b, b, 4.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.6e+53], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(N[(b + -2.0), $MachinePrecision] / N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + -16.0), $MachinePrecision]), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.6e53

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 80.8%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 80.2%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]
   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 80.2

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

   10. Applied egg-rr 80.2%

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

   if 3.6e53 < b < 1.35000000000000003e154

   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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 4.1

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

   8. Simplified 4.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

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

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

   10. Applied egg-rr 89.3%

       \[\leadsto \color{blue}{\frac{b + -2}{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)} \]

   if 1.35000000000000003e154 < 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. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-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 100.0

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

   11. Simplified 100.0%

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

Alternative 5: 72.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e+46)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1.35e+154)
     (* (/ (+ b -2.0) (fma b (* b (* b b)) -16.0)) (fma b b 4.0))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.75e+46) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.35e+154) {
		tmp = ((b + -2.0) / fma(b, (b * (b * b)), -16.0)) * fma(b, b, 4.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.75e+46)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(Float64(b + -2.0) / fma(b, Float64(b * Float64(b * b)), -16.0)) * fma(b, b, 4.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.75e+46], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(N[(b + -2.0), $MachinePrecision] / N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + -16.0), $MachinePrecision]), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.74999999999999992e46

   1. Initial program 98.4%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 81.8

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

   7. Simplified 81.8%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 71.4

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

   10. Simplified 71.4%

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

   if 1.74999999999999992e46 < b < 1.35000000000000003e154

   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

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 4.1

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

   8. Simplified 4.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

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

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

   10. Applied egg-rr 89.3%

       \[\leadsto \color{blue}{\frac{b + -2}{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)} \]

   if 1.35000000000000003e154 < 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. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-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 100.0

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

   11. Simplified 100.0%

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

Alternative 6: 73.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 700:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+99}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 700.0)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 2.05e+99)
     (* a (* a (* a -0.020833333333333332)))
     (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 700.0) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 2.05e+99) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 700.0)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 2.05e+99)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 700.0], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e+99], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 700

   1. Initial program 98.3%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 84.5

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

   7. Simplified 84.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.0

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

   10. Simplified 74.0%

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

   if 700 < b < 2.0499999999999999e99

   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 25.9%

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

   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.8

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

   8. Simplified 2.8%

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

   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. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 54.5

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

   11. Simplified 54.5%

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

   if 2.0499999999999999e99 < 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

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   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 95.3

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

   8. Simplified 95.3%

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

Alternative 7: 71.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 720:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 720.0)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1.35e+154)
     (* a (* a (* a -0.020833333333333332)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 720.0) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.35e+154) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 720.0)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.35e+154)
		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, 720.0], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 720

   1. Initial program 98.3%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 84.5

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

   7. Simplified 84.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.0

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

   10. Simplified 74.0%

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

   if 720 < b < 1.35000000000000003e154

   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 37.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   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. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 44.5

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

   11. Simplified 44.5%

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

   if 1.35000000000000003e154 < 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. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-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 100.0

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

   11. Simplified 100.0%

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

Alternative 8: 67.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 950:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a, -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 950.0)
   (/ 1.0 (fma a (fma 0.5 a -1.0) 2.0))
   (if (<= b 1.35e+154)
     (* a (* a (* a -0.020833333333333332)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 950.0) {
		tmp = 1.0 / fma(a, fma(0.5, a, -1.0), 2.0);
	} else if (b <= 1.35e+154) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 950.0)
		tmp = Float64(1.0 / fma(a, fma(0.5, a, -1.0), 2.0));
	elseif (b <= 1.35e+154)
		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, 950.0], N[(1.0 / N[(a * N[(0.5 * a + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 950

   1. Initial program 98.3%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 84.5

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

   7. Simplified 84.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 71.3

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

   10. Simplified 71.3%

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

   if 950 < b < 1.35000000000000003e154

   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 37.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   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. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 44.5

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

   11. Simplified 44.5%

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

   if 1.35000000000000003e154 < 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. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-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 100.0

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

   11. Simplified 100.0%

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

Alternative 9: 59.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 540:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 540.0)
   (/ 1.0 (- 2.0 a))
   (if (<= b 1.35e+154)
     (* a (* a (* a -0.020833333333333332)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 540.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 540.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 1.35d+154) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 540.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 540.0:
		tmp = 1.0 / (2.0 - a)
	elif b <= 1.35e+154:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 540.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 1.35e+154)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 540.0)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 1.35e+154)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 540.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 540

   1. Initial program 98.3%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 84.5

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

   7. Simplified 84.5%

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

   8. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 59.9

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

   10. Simplified 59.9%

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

   if 540 < b < 1.35000000000000003e154

   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 37.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   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. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 44.5

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

   11. Simplified 44.5%

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

   if 1.35000000000000003e154 < 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. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-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 100.0

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

   11. Simplified 100.0%

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

Alternative 10: 52.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 450:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 450.0) (/ 1.0 (- 2.0 a)) (* a (* a (* a -0.020833333333333332)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 450.0) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = a * (a * (a * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 450.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 450.0) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = a * (a * (a * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 450.0:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = a * (a * (a * -0.020833333333333332))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 450.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 450.0)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = a * (a * (a * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 450.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 450

   1. Initial program 98.3%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

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

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

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

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

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

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

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

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

      8. rec-exp N/A

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. neg-lowering-neg.f64 84.5

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

   7. Simplified 84.5%

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

   8. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 59.9

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

   10. Simplified 59.9%

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

   if 450 < b

   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 36.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   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. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   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. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 43.0

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

   11. Simplified 43.0%

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

Alternative 11: 40.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. clear-num N/A

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

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

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

   3. div-inv N/A

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

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

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

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

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

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

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

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

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

   8. rec-exp N/A

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

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

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

   10. neg-lowering-neg.f64 98.8

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. exp-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. neg-lowering-neg.f64 70.8

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

7. Simplified 70.8%

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

8. Taylor expanded in a around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 43.9

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

10. Simplified 43.9%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
11. Add Preprocessing

Alternative 12: 39.9% 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 81.8

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

5. Simplified 81.8%

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

6. Taylor expanded in b around 0

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

8. Simplified 43.2%

   \[\leadsto \color{blue}{0.5} \]
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 2024199 
(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))))