Quotient of sum of exps

Percentage Accurate: 99.0% → 99.0%

Time: 7.3s

Alternatives: 20

Speedup: 1.0×

• could not determine a ground truth

  1. a = 3.6536282858351007e+114
  2. b = 1.2711333217087346e-5

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Power[N[(N[Exp[(-a)], $MachinePrecision] * N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $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. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-exp.f64 N/A

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

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

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

   3. flip-+ N/A

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

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

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

   5. lower-/.f64 N/A

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

   6. sinh-cosh N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(e^{-a} - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.5e-7)
   (pow (- (exp (- a)) -1.0) -1.0)
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.5e-7) {
		tmp = pow((exp(-a) - -1.0), -1.0);
	} 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 (a <= (-7.5d-7)) then
        tmp = (exp(-a) - (-1.0d0)) ** (-1.0d0)
    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 (a <= -7.5e-7) {
		tmp = Math.pow((Math.exp(-a) - -1.0), -1.0);
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.5e-7:
		tmp = math.pow((math.exp(-a) - -1.0), -1.0)
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.5e-7)
		tmp = Float64(exp(Float64(-a)) - -1.0) ^ -1.0;
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.5e-7)
		tmp = (exp(-a) - -1.0) ^ -1.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.5e-7], N[Power[N[(N[Exp[(-a)], $MachinePrecision] - -1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5000000000000002e-7

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 98.9

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

   7. Applied rewrites 98.9%

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

   if -7.5000000000000002e-7 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(e^{-a} - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;\frac{e^{a}}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.4) (/ (exp a) (+ 2.0 a)) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.4) {
		tmp = exp(a) / (2.0 + a);
	} 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 (a <= (-0.4d0)) then
        tmp = exp(a) / (2.0d0 + a)
    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 (a <= -0.4) {
		tmp = Math.exp(a) / (2.0 + a);
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.4:
		tmp = math.exp(a) / (2.0 + a)
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.4)
		tmp = Float64(exp(a) / Float64(2.0 + a));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.4)
		tmp = exp(a) / (2.0 + a);
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.4], N[(N[Exp[a], $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

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

   8. Applied rewrites 98.9%

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

   if -0.40000000000000002 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;\frac{e^{a}}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.4) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.4) {
		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 (a <= (-0.4d0)) 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 (a <= -0.4) {
		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 a <= -0.4:
		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 (a <= -0.4)
		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 (a <= -0.4)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.4], 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 a < -0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

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

   if -0.40000000000000002 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{elif}\;a \leq -0.048:\\ \;\;\;\;{\left(\left(\left(\frac{\frac{2}{b} + 1}{b} + 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (pow (fma (- (* (* a a) -0.16666666666666666) 1.0) a 2.0) -1.0)
   (if (<= a -0.048)
     (pow (* (* (+ (/ (+ (/ 2.0 b) 1.0) b) 0.5) b) b) -1.0)
     (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = pow(fma((((a * a) * -0.16666666666666666) - 1.0), a, 2.0), -1.0);
	} else if (a <= -0.048) {
		tmp = pow(((((((2.0 / b) + 1.0) / b) + 0.5) * b) * b), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = fma(Float64(Float64(Float64(a * a) * -0.16666666666666666) - 1.0), a, 2.0) ^ -1.0;
	elseif (a <= -0.048)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(2.0 / b) + 1.0) / b) + 0.5) * b) * b) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[Power[N[(N[(N[(N[(a * a), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[a, -0.048], N[Power[N[(N[(N[(N[(N[(N[(2.0 / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e103

   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. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 100.0

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

   7. Applied rewrites 100.0%

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

   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 100.0%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 100.0%

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

   if -1e103 < a < -0.048000000000000001

   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. +-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 29.6

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

   5. Applied rewrites 29.6%

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

   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 13.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 52.6%

       \[\leadsto \frac{1}{\left(\left(\frac{\frac{2}{b} + 1}{b} + 0.5\right) \cdot b\right) \cdot b} \]

   if -0.048000000000000001 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   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 69.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{elif}\;a \leq -0.048:\\ \;\;\;\;{\left(\left(\left(\frac{\frac{2}{b} + 1}{b} + 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{elif}\;a \leq -112000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (pow (fma (- (* (* a a) -0.16666666666666666) 1.0) a 2.0) -1.0)
   (if (<= a -112000.0)
     (* (pow b 5.0) -0.0020833333333333333)
     (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = pow(fma((((a * a) * -0.16666666666666666) - 1.0), a, 2.0), -1.0);
	} else if (a <= -112000.0) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = fma(Float64(Float64(Float64(a * a) * -0.16666666666666666) - 1.0), a, 2.0) ^ -1.0;
	elseif (a <= -112000.0)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[Power[N[(N[(N[(N[(a * a), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[a, -112000.0], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e103

   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. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 100.0

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

   7. Applied rewrites 100.0%

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

   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 100.0%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 100.0%

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

   if -1e103 < a < -112000

   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 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

       \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   if -112000 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   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 68.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{elif}\;a \leq -112000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e+102)
   (pow (fma (- (* (fma -0.16666666666666666 a 0.5) a) 1.0) a 2.0) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+102) {
		tmp = pow(fma(((fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.2e+102)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.2e+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 + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.1999999999999995e102

   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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 77.3

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

   7. Applied rewrites 77.3%

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

   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 64.8%

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

   if 9.1999999999999995e102 < 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%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e+102)
   (pow (fma (- (* (* a a) -0.16666666666666666) 1.0) a 2.0) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+102) {
		tmp = pow(fma((((a * a) * -0.16666666666666666) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.2e+102)
		tmp = fma(Float64(Float64(Float64(a * a) * -0.16666666666666666) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.2e+102], N[Power[N[(N[(N[(N[(a * a), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.1999999999999995e102

   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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 77.3

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

   7. Applied rewrites 77.3%

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

   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 64.8%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 64.5%

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

   if 9.1999999999999995e102 < 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%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.16666666666666666 - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.7% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+103)
   (/ (exp a) 2.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+103) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+103)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+103], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e103

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{a}}} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.7%

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

   if 1e103 < 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%

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

4. Final simplification 78.9%

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

Alternative 12: 68.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+92}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 1\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+92)
   (pow (- (fma (- (* 0.5 a) 1.0) a 1.0) -1.0) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+92) {
		tmp = pow((fma(((0.5 * a) - 1.0), a, 1.0) - -1.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+92)
		tmp = Float64(fma(Float64(Float64(0.5 * a) - 1.0), a, 1.0) - -1.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+92], N[Power[N[(N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] - -1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0000000000000001e92

   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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 77.2

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

   7. Applied rewrites 77.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.9%

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

   if 2.0000000000000001e92 < 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 97.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+92}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 1\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 1\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.15e+140)
   (pow (- (fma (- (* 0.5 a) 1.0) a 1.0) -1.0) -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 <= 2.15e+140) {
		tmp = pow((fma(((0.5 * a) - 1.0), a, 1.0) - -1.0), -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 <= 2.15e+140)
		tmp = Float64(fma(Float64(Float64(0.5 * a) - 1.0), a, 1.0) - -1.0) ^ -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, 2.15e+140], N[Power[N[(N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] - -1.0), $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 < 2.15000000000000001e140

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 60.8%

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

   if 2.15000000000000001e140 < 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 95.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 1\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.85:\\ \;\;\;\;{\left(\left(1 - a\right) - -1\right)}^{-1}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\left(a \cdot a\right) \cdot 0.5\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.85)
   (pow (- (- 1.0 a) -1.0) -1.0)
   (if (<= b 2.15e+140)
     (pow (* (* a a) 0.5) -1.0)
     (pow (* (* 0.5 b) b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.85) {
		tmp = pow(((1.0 - a) - -1.0), -1.0);
	} else if (b <= 2.15e+140) {
		tmp = pow(((a * a) * 0.5), -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 <= 0.85d0) then
        tmp = ((1.0d0 - a) - (-1.0d0)) ** (-1.0d0)
    else if (b <= 2.15d+140) then
        tmp = ((a * a) * 0.5d0) ** (-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 <= 0.85) {
		tmp = Math.pow(((1.0 - a) - -1.0), -1.0);
	} else if (b <= 2.15e+140) {
		tmp = Math.pow(((a * a) * 0.5), -1.0);
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.85:
		tmp = math.pow(((1.0 - a) - -1.0), -1.0)
	elif b <= 2.15e+140:
		tmp = math.pow(((a * a) * 0.5), -1.0)
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.85)
		tmp = Float64(Float64(1.0 - a) - -1.0) ^ -1.0;
	elseif (b <= 2.15e+140)
		tmp = Float64(Float64(a * a) * 0.5) ^ -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 <= 0.85)
		tmp = ((1.0 - a) - -1.0) ^ -1.0;
	elseif (b <= 2.15e+140)
		tmp = ((a * a) * 0.5) ^ -1.0;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.85], N[Power[N[(N[(1.0 - a), $MachinePrecision] - -1.0), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 2.15e+140], N[Power[N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 0.849999999999999978

   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. 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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 54.0%

       \[\leadsto \frac{1}{\left(1 - a\right) - -1} \]

   if 0.849999999999999978 < b < 2.15000000000000001e140

   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. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 53.5

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

   7. Applied rewrites 53.5%

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

   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 34.4%

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

   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 33.9%

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

   if 2.15000000000000001e140 < 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 95.2%

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

   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 95.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.85:\\ \;\;\;\;{\left(\left(1 - a\right) - -1\right)}^{-1}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\left(a \cdot a\right) \cdot 0.5\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.15e+140)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -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 <= 2.15e+140) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -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 <= 2.15e+140)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -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, 2.15e+140], 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 + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.15000000000000001e140

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 76.5

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

   7. Applied rewrites 76.5%

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

   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 60.8%

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

   if 2.15000000000000001e140 < 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 95.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+140}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+141}:\\ \;\;\;\;{\left(\left(a \cdot a\right) \cdot 0.5\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.4e+141)
   (pow (* (* a a) 0.5) -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 (a <= -3.4e+141) {
		tmp = pow(((a * a) * 0.5), -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 (a <= -3.4e+141)
		tmp = Float64(Float64(a * a) * 0.5) ^ -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[a, -3.4e+141], N[Power[N[(N[(a * a), $MachinePrecision] * 0.5), $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 a < -3.3999999999999998e141

   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. 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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 100.0

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

   7. Applied rewrites 100.0%

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

   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 92.5%

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

   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 92.5%

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

   if -3.3999999999999998e141 < a

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. +-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 87.1

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

   5. Applied rewrites 87.1%

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

   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 58.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+141}:\\ \;\;\;\;{\left(\left(a \cdot a\right) \cdot 0.5\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+89}:\\ \;\;\;\;{\left(\left(1 - a\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.5e+89) (pow (- (- 1.0 a) -1.0) -1.0) (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+89) {
		tmp = pow(((1.0 - a) - -1.0), -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 <= 8.5d+89) then
        tmp = ((1.0d0 - a) - (-1.0d0)) ** (-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 <= 8.5e+89) {
		tmp = Math.pow(((1.0 - a) - -1.0), -1.0);
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.5e+89:
		tmp = math.pow(((1.0 - a) - -1.0), -1.0)
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.5e+89)
		tmp = Float64(Float64(1.0 - a) - -1.0) ^ -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 <= 8.5e+89)
		tmp = ((1.0 - a) - -1.0) ^ -1.0;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.5e+89], N[Power[N[(N[(1.0 - a), $MachinePrecision] - -1.0), $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 < 8.50000000000000045e89

   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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. exp-neg N/A

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

      6. lft-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 77.2

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

   7. Applied rewrites 77.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 49.6%

       \[\leadsto \frac{1}{\left(1 - a\right) - -1} \]

   if 8.50000000000000045e89 < 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 83.2%

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

   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 83.2%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+89}:\\ \;\;\;\;{\left(\left(1 - a\right) - -1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[Power[N[(N[(1.0 - a), $MachinePrecision] - -1.0), $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-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

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

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

   5. exp-neg N/A

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

   6. lft-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 69.0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 41.7%

    \[\leadsto \frac{1}{\left(1 - a\right) - -1} \]

11. Final simplification 41.7%

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

Alternative 19: 40.7% 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-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

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

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

   5. exp-neg N/A

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

   6. lft-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 69.0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 41.7%

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

11. Final simplification 41.7%

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

Alternative 20: 39.9% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. 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 79.9

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

5. Applied rewrites 79.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 40.9%

   \[\leadsto 0.5 \]
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 2024342 
(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))))