Quotient of sum of exps

Percentage Accurate: 98.8% → 99.4%

Time: 6.9s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

  1. a = 1.3097929029970875e+118
  2. b = -6.44518245457665e-211

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: 98.8% 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: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-1 - \frac{e^{b}}{e^{a}}} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return -1.0 / (-1.0 - (exp(b) / exp(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) / exp(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-*r/ 100.0%

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

   7. exp-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-frac-neg2 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. sub-neg 100.0%

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

   14. *-commutative 100.0%

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

   15. sub-neg 100.0%

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

   16. distribute-neg-in 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -880:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -880.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -880.0) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-880.0d0)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -880.0) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -880.0:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -880.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -880.0)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -880.0], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -880

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   if -880 < a

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -880:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e+95)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (* a -0.16666666666666666))))))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.1e+95) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.1d+95)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (a * (-0.16666666666666666d0))))))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.1e+95) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.1e+95:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.1e+95)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(a * -0.16666666666666666))))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.1e+95)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.1e+95], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1000000000000003e95

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 91.2%

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

   9. Taylor expanded in a around inf 91.2%

      \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(-0.16666666666666666 \cdot a\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. *-commutative 91.2%

          \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.16666666666666666\right)} - 1\right)} \]

   11. Simplified 91.2%

       \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.16666666666666666\right)} - 1\right)} \]

   if -3.1000000000000003e95 < a

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 95.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 2.9× 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]

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. remove-double-neg 100.0%

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

   5. unsub-neg 100.0%

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

   6. div-sub 76.2%

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

   7. *-lft-identity 76.2%

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

   8. associate-*l/ 76.2%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 72.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot 0.16666666666666666\right)\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - t\_0 \cdot t\_0}{b \cdot 0.5 - t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b 0.16666666666666666))))
   (if (<= b 2.8e+77)
     (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
     (if (<= b 2e+154)
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (/ (- (* (* b 0.5) (* b 0.5)) (* t_0 t_0)) (- (* b 0.5) t_0))))))
       (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * 0.16666666666666666);
	double tmp;
	if (b <= 2.8e+77) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 2e+154) {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * 0.16666666666666666d0)
    if (b <= 2.8d+77) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 2d+154) then
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((((b * 0.5d0) * (b * 0.5d0)) - (t_0 * t_0)) / ((b * 0.5d0) - t_0)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * 0.16666666666666666);
	double tmp;
	if (b <= 2.8e+77) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 2e+154) {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * 0.16666666666666666)
	tmp = 0
	if b <= 2.8e+77:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 2e+154:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * 0.16666666666666666))
	tmp = 0.0
	if (b <= 2.8e+77)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 2e+154)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(Float64(Float64(b * 0.5) * Float64(b * 0.5)) - Float64(t_0 * t_0)) / Float64(Float64(b * 0.5) - t_0))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * 0.16666666666666666);
	tmp = 0.0;
	if (b <= 2.8e+77)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 2e+154)
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e+77], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+154], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(N[(N[(b * 0.5), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e77

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 72.4%

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

      1. rec-exp 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in a around 0 65.9%

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

   if 2.8e77 < b < 2.00000000000000007e154

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 64.2%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 64.2%

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

      1. distribute-lft-in 64.2%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(b \cdot 0.5 + b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\right)} \]

      2. flip-+ 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 2.00000000000000007e154 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right) \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}{b \cdot 0.5 - b \cdot \left(b \cdot 0.16666666666666666\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.45 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.45e+99)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.45e+99) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.45d+99) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.45e+99) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.45e+99:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.45e+99)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.45e+99)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.45e+99], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4499999999999998e99

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 70.7%

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

      1. rec-exp 70.6%

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

   7. Simplified 70.6%

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

   8. Taylor expanded in a around 0 64.3%

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

   if 2.4499999999999998e99 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 98.4%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 98.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.45 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e+127)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e+127) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.2d+127) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.2e+127) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.2e+127:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.2e+127)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.2e+127)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.2e+127], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002e127

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 69.8%

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

      1. rec-exp 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in a around 0 63.7%

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

   if 2.2000000000000002e127 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 85.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e+128)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (* a -0.16666666666666666))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e+128) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9.5d+128) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (a * (-0.16666666666666666d0))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e+128) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e+128:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e+128)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(a * -0.16666666666666666))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e+128)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (a * -0.16666666666666666)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e+128], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.50000000000000014e128

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 69.8%

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

      1. rec-exp 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in a around 0 63.7%

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

   9. Taylor expanded in a around inf 63.4%

      \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(-0.16666666666666666 \cdot a\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. *-commutative 63.4%

          \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.16666666666666666\right)} - 1\right)} \]

   11. Simplified 63.4%

       \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.16666666666666666\right)} - 1\right)} \]

   if 9.50000000000000014e128 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 85.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.2e+128)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+128) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.2d+128) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.2e+128) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.2e+128:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.2e+128)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.2e+128)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.2e+128], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2000000000000001e128

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 69.8%

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

      1. rec-exp 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in a around 0 59.1%

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

   if 1.2000000000000001e128 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 85.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.9% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5))))))

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Python

def code(a, b):
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-*r/ 100.0%

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

   7. exp-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-frac-neg2 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. sub-neg 100.0%

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

   14. *-commutative 100.0%

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

   15. sub-neg 100.0%

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

   16. distribute-neg-in 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.4%

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

   1. rec-exp 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in a around 0 51.7%

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

9. Final simplification 51.7%

   \[\leadsto \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 11: 40.1% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-*r/ 100.0%

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

   7. exp-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-frac-neg2 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. sub-neg 100.0%

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

   14. *-commutative 100.0%

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

   15. sub-neg 100.0%

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

   16. distribute-neg-in 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.4%

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

   1. rec-exp 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in a around 0 38.6%

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

   1. neg-mul-1 38.6%

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

   2. unsub-neg 38.6%

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

10. Simplified 38.6%

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

Alternative 12: 39.4% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-*r/ 100.0%

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

   7. exp-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-frac-neg2 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. sub-neg 100.0%

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

   14. *-commutative 100.0%

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

   15. sub-neg 100.0%

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

   16. distribute-neg-in 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.4%

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

   1. rec-exp 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in a around 0 37.9%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
9. Step-by-step derivation

   1. *-commutative 37.9%

      \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

10. Simplified 37.9%

    \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
11. Add Preprocessing

Alternative 13: 39.3% accurate, 305.0× 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 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-*r/ 100.0%

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

   7. exp-neg 100.0%

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

   8. distribute-neg-frac 100.0%

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

   9. distribute-frac-neg2 100.0%

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

   10. distribute-lft-neg-out 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. sub-neg 100.0%

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

   14. *-commutative 100.0%

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

   15. sub-neg 100.0%

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

   16. distribute-neg-in 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - \frac{e^{b}}{e^{a}}}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 85.4%

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

6. Taylor expanded in b around 0 37.6%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.9× 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 2024172 
(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))))