Quotient of sum of exps

Percentage Accurate: 98.9% → 98.5%

Time: 6.5s

Alternatives: 13

Speedup: 1.4×

• could not determine a ground truth

  1. a = 5.383688734304513e+229
  2. b = 6.921328750558864e+234

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-11} \lor \neg \left(b \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -2.4e-11) (not (<= b 8.8e-13)))
   (pow (+ (exp b) 1.0) -1.0)
   (/ (exp a) (+ (exp a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -2.4e-11) || !(b <= 8.8e-13)) {
		tmp = pow((exp(b) + 1.0), -1.0);
	} else {
		tmp = exp(a) / (exp(a) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-2.4d-11)) .or. (.not. (b <= 8.8d-13))) then
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    else
        tmp = exp(a) / (exp(a) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -2.4e-11) || !(b <= 8.8e-13)) {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	} else {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -2.4e-11) or not (b <= 8.8e-13):
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	else:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -2.4e-11) || !(b <= 8.8e-13))
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	else
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -2.4e-11) || ~((b <= 8.8e-13)))
		tmp = (exp(b) + 1.0) ^ -1.0;
	else
		tmp = exp(a) / (exp(a) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -2.4e-11], N[Not[LessEqual[b, 8.8e-13]], $MachinePrecision]], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4000000000000001e-11 or 8.79999999999999986e-13 < b

   1. Initial program 99.2%

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

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

   if -2.4000000000000001e-11 < b < 8.79999999999999986e-13

   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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-11} \lor \neg \left(b \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-11} \lor \neg \left(b \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-a} - -1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -2.4e-11) (not (<= b 8.8e-13)))
   (pow (+ (exp b) 1.0) -1.0)
   (pow (- (exp (- a)) -1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -2.4e-11) || !(b <= 8.8e-13)) {
		tmp = pow((exp(b) + 1.0), -1.0);
	} else {
		tmp = pow((exp(-a) - -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 ((b <= (-2.4d-11)) .or. (.not. (b <= 8.8d-13))) then
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    else
        tmp = (exp(-a) - (-1.0d0)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -2.4e-11) || !(b <= 8.8e-13)) {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	} else {
		tmp = Math.pow((Math.exp(-a) - -1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -2.4e-11) or not (b <= 8.8e-13):
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	else:
		tmp = math.pow((math.exp(-a) - -1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -2.4e-11) || !(b <= 8.8e-13))
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	else
		tmp = Float64(exp(Float64(-a)) - -1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -2.4e-11) || ~((b <= 8.8e-13)))
		tmp = (exp(b) + 1.0) ^ -1.0;
	else
		tmp = (exp(-a) - -1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -2.4e-11], N[Not[LessEqual[b, 8.8e-13]], $MachinePrecision]], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[Exp[(-a)], $MachinePrecision] - -1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4000000000000001e-11 or 8.79999999999999986e-13 < b

   1. Initial program 99.2%

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

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

   if -2.4000000000000001e-11 < b < 8.79999999999999986e-13

   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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

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

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

      3. *-rgt-identity 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 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-11} \lor \neg \left(b \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-a} - -1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.66:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.66) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.66) {
		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 <= (-0.66d0)) 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 <= -0.66) {
		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 <= -0.66:
		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 <= -0.66)
		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 <= -0.66)
		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, -0.66], 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 < -0.660000000000000031

   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 98.1%

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

   if -0.660000000000000031 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. 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 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.66:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+97}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e+97)
   (pow (fma (- (* (fma -0.16666666666666666 a 0.5) a) 1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e+97) {
		tmp = pow(fma(((fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e+97)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e+97], N[Power[N[(N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000001e97

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

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

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

      3. *-rgt-identity 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 71.8

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

   7. Applied rewrites 71.8%

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

   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 61.8%

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

   if 3.70000000000000001e97 < b

   1. Initial program 97.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 97.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 97.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+97}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.6e+98)
   (/ (exp a) 2.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.6e+98) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.6e+98)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.6e+98], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999981e98

   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.3%

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

   if 3.59999999999999981e98 < b

   1. Initial program 97.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 97.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 97.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e+97)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.05e+97) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.05e+97)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e+97], N[Power[N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05000000000000006e97

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

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

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

      3. *-rgt-identity 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 71.7

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

   7. Applied rewrites 71.7%

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

   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 58.6%

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

   if 1.05000000000000006e97 < b

   1. Initial program 97.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 95.6%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 95.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+139)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (* (fma 0.5 b 1.0) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+139) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow((fma(0.5, b, 1.0) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+139)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(fma(0.5, b, 1.0) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+139], 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), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999978e139

   1. Initial program 99.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.3

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. distribute-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. fp-cancel-sign-sub-inv N/A

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

      3. *-rgt-identity 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 69.6

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

   7. Applied rewrites 69.6%

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

   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 56.7%

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

   if 3.49999999999999978e139 < 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 85.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 85.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-47}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-47)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-47) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-47)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-47], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000002e-47

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

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

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

      3. *-rgt-identity 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 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.5%

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

   if 1.65000000000000002e-47 < b

   1. Initial program 98.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   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 43.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-47}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+27}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.15e+27) (pow (- 2.0 a) -1.0) (pow (* (fma 0.5 b 1.0) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.15e+27) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow((fma(0.5, b, 1.0) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.15e+27)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(fma(0.5, b, 1.0) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.15e+27], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.15000000000000004e27

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

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

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

      3. *-rgt-identity 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 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 49.5%

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

   if 2.15000000000000004e27 < b

   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 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 46.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 46.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+27}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+27}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.15e+27) (pow (- 2.0 a) -1.0) (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.15e+27) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.15d+27) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((0.5d0 * b) * b) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.15e+27) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.15e+27:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.15e+27)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.15e+27)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.15e+27], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.15000000000000004e27

   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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

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

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

      3. *-rgt-identity 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 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 49.5%

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

   if 2.15000000000000004e27 < b

   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 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 46.8%

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

   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 46.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+27}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 99.4

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

   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 99.4

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

4. Applied rewrites 99.4%

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

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

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

   3. *-rgt-identity 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 64.5

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 39.1%

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

11. Final simplification 39.1%

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

Alternative 13: 39.6% 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 38.1%

   \[\leadsto 0.5 \]
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 2024337 
(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))))