Quotient of sum of exps

Percentage Accurate: 98.9% → 98.6%

Time: 5.4s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.8743764188719624e+97
  2. b = -1.5862026759655962e-73

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{a}}{2} \]

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.9%

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

Alternative 2: 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 3: 70.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e+102)
   (pow (fma (- (* (fma -0.16666666666666666 a 0.5) a) 1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+102) {
		tmp = pow(fma(((fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e+102)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e+102], N[Power[N[(N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.1999999999999999e102

   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. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 71.4

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

   7. Applied rewrites 71.4%

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

   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 62.8%

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

   if 3.1999999999999999e102 < 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e+103)
   (/ (exp a) 2.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e+103], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0500000000000001e103

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.6%

      \[\leadsto \frac{e^{a}}{2} \]

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 74.1%

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

Alternative 5: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e+102)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e+102) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e+102)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e+102], N[Power[N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.1999999999999999e102

   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. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 71.4

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

   7. Applied rewrites 71.4%

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

   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 57.6%

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

   if 3.1999999999999999e102 < 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e+118)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+118) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e+118)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e+118], N[Power[N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000016e118

   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. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 70.9

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

   7. Applied rewrites 70.9%

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

   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 57.3%

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

   if 3.80000000000000016e118 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 86.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-25)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e-25) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e-25)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e-25], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4999999999999999e-25

   1. Initial program 98.3%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 77.6

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

   7. Applied rewrites 77.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 50.9%

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

   if 1.4999999999999999e-25 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 44.2%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.6e+44) (pow (- 2.0 a) -1.0) (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.6e+44) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((0.5 * b) * 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 (b <= 2.6d+44) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((0.5d0 * b) * b) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.6e+44) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.6e+44:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.6e+44)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.6e+44)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.6e+44], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5999999999999999e44

   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. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 74.2

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.4%

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

   if 2.5999999999999999e44 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 53.4%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

   5. sinh-cosh N/A

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

   6. sinh-cosh N/A

      \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

   7. sinh---cosh-rev N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f64 N/A

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

   10. sinh-cosh N/A

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

   11. lower-*.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 98.8

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in b around 0

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

   1. distribute-rgt-in N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. exp-neg N/A

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

   6. rgt-mult-inverse N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 66.2

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

7. Applied rewrites 66.2%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 37.3%

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

11. Final simplification 37.3%

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

Alternative 10: 38.8% 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. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 80.7

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

5. Applied rewrites 80.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.4%

   \[\leadsto 0.5 \]
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 2024326 
(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))))