Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 7.7s

Alternatives: 20

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.5115879800628288e+248
  2. b = 9.391135181021694e-119

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: 98.9% 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 2: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.001:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.001) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.001) {
		tmp = 1.0 / (1.0 + exp(-a));
	} 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) <= 0.001d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    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) <= 0.001) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.001:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.001)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	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) <= 0.001)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.001], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1e-3

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.7

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 1e-3 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.001:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 97.9%

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

Alternative 4: 93.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right) \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, -1\right)\right), -4\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (/ 1.0 (* 0.5 (* a a)))
   (if (<= a -4.2e+36)
     (/
      1.0
      (/
       (fma a (* (fma a 0.5 -1.0) (* a (fma a 0.5 -1.0))) -4.0)
       (fma a (fma a 0.5 -1.0) -2.0)))
     (/ 1.0 (+ (exp b) 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = 1.0 / (0.5 * (a * a));
	} else if (a <= -4.2e+36) {
		tmp = 1.0 / (fma(a, (fma(a, 0.5, -1.0) * (a * fma(a, 0.5, -1.0))), -4.0) / fma(a, fma(a, 0.5, -1.0), -2.0));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	elseif (a <= -4.2e+36)
		tmp = Float64(1.0 / Float64(fma(a, Float64(fma(a, 0.5, -1.0) * Float64(a * fma(a, 0.5, -1.0))), -4.0) / fma(a, fma(a, 0.5, -1.0), -2.0)));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+154], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.2e+36], N[(1.0 / N[(N[(a * N[(N[(a * 0.5 + -1.0), $MachinePrecision] * N[(a * N[(a * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision] / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 100.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{1}{0.5 \cdot \left(a \cdot \color{blue}{a}\right)} \]

   if -2.00000000000000007e154 < a < -4.20000000000000009e36

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 96.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 5.9%

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

   12. Applied rewrites 77.0%

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

   if -4.20000000000000009e36 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right) \cdot \left(a \cdot \mathsf{fma}\left(a, 0.5, -1\right)\right), -4\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq 1.55 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, t\_0 \cdot t\_0, -4\right)}{\mathsf{fma}\left(b, t\_0, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b 1.55e-26)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (if (<= b 1.05e+103)
       (/ 1.0 (/ (fma (* b b) (* t_0 t_0) -4.0) (fma b t_0 -2.0)))
       (/ 1.0 (* b (* (* b b) 0.16666666666666666)))))))

C

double code(double a, double b) {
	double t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (b <= 1.55e-26) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.05e+103) {
		tmp = 1.0 / (fma((b * b), (t_0 * t_0), -4.0) / fma(b, t_0, -2.0));
	} else {
		tmp = 1.0 / (b * ((b * b) * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (b <= 1.55e-26)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.05e+103)
		tmp = Float64(1.0 / Float64(fma(Float64(b * b), Float64(t_0 * t_0), -4.0) / fma(b, t_0, -2.0)));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(b * b) * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 1.55e-26], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -4.0), $MachinePrecision] / N[(b * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.54999999999999992e-26

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.7

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 80.7

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

   7. Applied rewrites 80.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 69.5%

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

   if 1.54999999999999992e-26 < b < 1.0500000000000001e103

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 19.8%

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

   10. Applied rewrites 82.4%

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

   if 1.0500000000000001e103 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -4\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.1e+57)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1e+154)
     (/
      1.0
      (fma
       (*
        b
        (fma
         (fma b 0.16666666666666666 0.5)
         (* (* b b) (fma b 0.16666666666666666 0.5))
         -1.0))
       (/ 1.0 (fma b (fma b 0.16666666666666666 0.5) -1.0))
       2.0))
     (/ 1.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.1e+57) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1e+154) {
		tmp = 1.0 / fma((b * fma(fma(b, 0.16666666666666666, 0.5), ((b * b) * fma(b, 0.16666666666666666, 0.5)), -1.0)), (1.0 / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.1e+57)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1e+154)
		tmp = Float64(1.0 / fma(Float64(b * fma(fma(b, 0.16666666666666666, 0.5), Float64(Float64(b * b) * fma(b, 0.16666666666666666, 0.5)), -1.0)), Float64(1.0 / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.1e+57], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+154], N[(1.0 / N[(N[(b * N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.09999999999999975e57

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.8

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.7%

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

   if 6.09999999999999975e57 < b < 1.00000000000000004e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 38.9%

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

   10. Applied rewrites 96.0%

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

   if 1.00000000000000004e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.1e+57)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1e+154)
     (/
      1.0
      (fma
       b
       (/
        (fma
         (fma b 0.16666666666666666 0.5)
         (* (* b b) (fma b 0.16666666666666666 0.5))
         -1.0)
        (fma b (fma b 0.16666666666666666 0.5) -1.0))
       2.0))
     (/ 1.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.1e+57) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1e+154) {
		tmp = 1.0 / fma(b, (fma(fma(b, 0.16666666666666666, 0.5), ((b * b) * fma(b, 0.16666666666666666, 0.5)), -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.1e+57)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1e+154)
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(b, 0.16666666666666666, 0.5), Float64(Float64(b * b) * fma(b, 0.16666666666666666, 0.5)), -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.1e+57], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+154], N[(1.0 / N[(b * N[(N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.09999999999999975e57

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.8

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.7%

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

   if 6.09999999999999975e57 < b < 1.00000000000000004e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 38.9%

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

   10. Applied rewrites 88.2%

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

   if 1.00000000000000004e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.1e+57)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1.9e+154)
     (/
      1.0
      (/
       (fma (fma b 0.5 1.0) (* b (fma b (* b 0.5) b)) -4.0)
       (fma b (fma b 0.5 1.0) -2.0)))
     (/ 1.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.1e+57) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.9e+154) {
		tmp = 1.0 / (fma(fma(b, 0.5, 1.0), (b * fma(b, (b * 0.5), b)), -4.0) / fma(b, fma(b, 0.5, 1.0), -2.0));
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.1e+57)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.9e+154)
		tmp = Float64(1.0 / Float64(fma(fma(b, 0.5, 1.0), Float64(b * fma(b, Float64(b * 0.5), b)), -4.0) / fma(b, fma(b, 0.5, 1.0), -2.0)));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.1e+57], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e+154], N[(1.0 / N[(N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision] / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.09999999999999975e57

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.8

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.7%

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

   if 6.09999999999999975e57 < b < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 88.1%

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

   if 1.8999999999999999e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

Alternative 9: 72.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{0.25 \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.1e+57)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1.9e+154)
     (/ 1.0 (/ (* 0.25 (* (* b b) (* b b))) (fma b (fma b 0.5 1.0) -2.0)))
     (/ 1.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.1e+57) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.9e+154) {
		tmp = 1.0 / ((0.25 * ((b * b) * (b * b))) / fma(b, fma(b, 0.5, 1.0), -2.0));
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.1e+57)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.9e+154)
		tmp = Float64(1.0 / Float64(Float64(0.25 * Float64(Float64(b * b) * Float64(b * b))) / fma(b, fma(b, 0.5, 1.0), -2.0)));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.1e+57], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e+154], N[(1.0 / N[(N[(0.25 * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.09999999999999975e57

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.8

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.7%

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

   if 6.09999999999999975e57 < b < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 88.1%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 88.1%

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

   if 1.8999999999999999e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

Alternative 10: 70.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{+93}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot \left(b \cdot b\right), 0.16666666666666666, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.02e+93)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/
    1.0
    (fma (* b (* b b)) 0.16666666666666666 (fma b (fma b 0.5 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.02e+93) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma((b * (b * b)), 0.16666666666666666, fma(b, fma(b, 0.5, 1.0), 2.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.02e+93)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(Float64(b * Float64(b * b)), 0.16666666666666666, fma(b, fma(b, 0.5, 1.0), 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.02e+93], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0200000000000001e93

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.9

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.0%

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

   if 1.0200000000000001e93 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.5%

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

   10. Applied rewrites 91.5%

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

   12. Applied rewrites 93.5%

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

Alternative 11: 70.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 9.4999999999999992e102

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 74.9

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 64.3%

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

   if 9.4999999999999992e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 69.4%

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

Alternative 12: 54.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq -19:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot 0.5, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, a \cdot -0.020833333333333332, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e+120)
   (/ 1.0 (* 0.5 (* a a)))
   (if (<= a -19.0)
     (/ 1.0 (fma b (* b 0.5) b))
     (fma a (fma a (* a -0.020833333333333332) 0.25) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e+120) {
		tmp = 1.0 / (0.5 * (a * a));
	} else if (a <= -19.0) {
		tmp = 1.0 / fma(b, (b * 0.5), b);
	} else {
		tmp = fma(a, fma(a, (a * -0.020833333333333332), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e+120)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	elseif (a <= -19.0)
		tmp = Float64(1.0 / fma(b, Float64(b * 0.5), b));
	else
		tmp = fma(a, fma(a, Float64(a * -0.020833333333333332), 0.25), 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e+120], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -19.0], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.80000000000000008e120

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 100.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 86.3%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 86.3%

       \[\leadsto \frac{1}{0.5 \cdot \left(a \cdot \color{blue}{a}\right)} \]

   if -1.80000000000000008e120 < a < -19

   1. Initial program 96.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 26.1%

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

   if -19 < a

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.1

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.6%

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, a \cdot -0.020833333333333332, 0.25\right)}, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq -18:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, a \cdot -0.020833333333333332, 0.25\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e+120)
   (/ 1.0 (* 0.5 (* a a)))
   (if (<= a -18.0)
     (/ 1.0 (* b (* b 0.5)))
     (fma a (fma a (* a -0.020833333333333332) 0.25) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e+120) {
		tmp = 1.0 / (0.5 * (a * a));
	} else if (a <= -18.0) {
		tmp = 1.0 / (b * (b * 0.5));
	} else {
		tmp = fma(a, fma(a, (a * -0.020833333333333332), 0.25), 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e+120)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	elseif (a <= -18.0)
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	else
		tmp = fma(a, fma(a, Float64(a * -0.020833333333333332), 0.25), 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e+120], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -18.0], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.80000000000000008e120

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 100.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 86.3%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 86.3%

       \[\leadsto \frac{1}{0.5 \cdot \left(a \cdot \color{blue}{a}\right)} \]

   if -1.80000000000000008e120 < a < -18

   1. Initial program 96.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 25.9%

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

   if -18 < a

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.1

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.6%

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, a \cdot -0.020833333333333332, 0.25\right)}, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.2e+102)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (* b (* (* b b) 0.16666666666666666)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.2e+102) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * ((b * b) * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.2e+102)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(b * b) * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.2e+102], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999999e102

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 74.9

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 59.9%

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

   if 8.1999999999999999e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.55e+151)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (* b (* b 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.55e+151) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.55e+151)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.55e+151], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5500000000000001e151

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 98.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.8%

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

   if 1.5500000000000001e151 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

Alternative 16: 60.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e+120)
   (/ 1.0 (* 0.5 (* a a)))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e+120) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e+120)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e+120], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.80000000000000008e120

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 100.0

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 86.3%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 86.3%

       \[\leadsto \frac{1}{0.5 \cdot \left(a \cdot \color{blue}{a}\right)} \]

   if -1.80000000000000008e120 < 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. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.5%

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

Alternative 17: 53.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95) (/ 1.0 (- 2.0 a)) (/ 1.0 (* b (* b 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (b * (b * 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 (b <= 1.95d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = 1.0d0 / (b * (b * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.95) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.95:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 1.0 / (b * (b * 0.5))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.95)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 1.0 / (b * (b * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.94999999999999996

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. metadata-eval 97.7

          \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. exp-neg N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 80.3

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 54.2%

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

   if 1.94999999999999996 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.1%

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

Alternative 18: 40.1% 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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. rec-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. metadata-eval 98.2

       \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. exp-neg N/A

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

   5. lft-mult-inverse N/A

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

   6. *-rgt-identity N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 72.4

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

7. Applied rewrites 72.4%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 42.5%

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

Alternative 19: 39.4% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, 0.25, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma a 0.25 0.5))

C

double code(double a, double b) {
	return fma(a, 0.25, 0.5);
}

Julia

function code(a, b)
	return fma(a, 0.25, 0.5)
end

Wolfram

code[a_, b_] := N[(a * 0.25 + 0.5), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. div-inv N/A

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

   9. lower-*.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. rec-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. metadata-eval 98.2

       \[\leadsto {\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{{\left({\left(\left(e^{a} + e^{b}\right) \cdot e^{-a}\right)}^{-0.5}\right)}^{2}} \]

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. exp-neg N/A

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

   5. lft-mult-inverse N/A

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

   6. *-rgt-identity N/A

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

   7. lower-+.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 72.4

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

7. Applied rewrites 72.4%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 41.7%

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

Alternative 20: 39.2% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 41.7%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024234 
(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))))