Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 7.1s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% 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: 100.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / ((2.0 + math.exp((b - a))) + -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 76.9%

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

   7. *-lft-identity 76.9%

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

   8. associate-*l/ 76.9%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 99.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{b - a}\right)\right)}} \]

   2. expm1-undefine 99.4%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} - 1}} \]

   3. div-exp 99.0%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(1 + \color{blue}{\frac{e^{b}}{e^{a}}}\right)} - 1} \]

   4. +-commutative 99.0%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\color{blue}{\frac{e^{b}}{e^{a}} + 1}\right)} - 1} \]

   5. div-exp 99.4%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\color{blue}{e^{b - a}} + 1\right)} - 1} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(e^{b - a} + 1\right)} - 1}} \]
7. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(e^{b - a} + 1\right)} + \left(-1\right)}} \]

   2. log1p-undefine 99.1%

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

   3. rem-exp-log 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-+r+ 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e-9) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e-9) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8.6d-9)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.6e-9) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e-9:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e-9)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.6e-9)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e-9], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.59999999999999925e-9

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. associate-*r/ 99.9%

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

      6. exp-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. distribute-frac-neg2 99.9%

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

      10. distribute-lft-neg-out 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. sub-neg 99.9%

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

      14. *-commutative 99.9%

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

      15. sub-neg 99.9%

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

      16. distribute-neg-in 99.9%

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

   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 -8.59999999999999925e-9 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. associate-*r/ 99.4%

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

      6. exp-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. distribute-frac-neg2 99.4%

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

      10. distribute-lft-neg-out 99.4%

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

      11. +-commutative 99.4%

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

      12. distribute-neg-in 99.4%

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

      13. sub-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. sub-neg 99.4%

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

      16. distribute-neg-in 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 98.8%

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

Alternative 3: 98.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.02e-8) (/ (exp a) (+ 2.0 a)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.02e-8) {
		tmp = exp(a) / (2.0 + a);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.02d-8)) then
        tmp = exp(a) / (2.0d0 + a)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.02e-8) {
		tmp = Math.exp(a) / (2.0 + a);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.02e-8:
		tmp = math.exp(a) / (2.0 + a)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.02e-8)
		tmp = Float64(exp(a) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.02e-8)
		tmp = exp(a) / (2.0 + a);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.02e-8], N[(N[Exp[a], $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.02000000000000003e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0 99.9%

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

   4. Taylor expanded in a around 0 99.0%

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

      1. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   if -1.02000000000000003e-8 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. associate-*r/ 99.4%

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

      6. exp-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. distribute-frac-neg2 99.4%

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

      10. distribute-lft-neg-out 99.4%

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

      11. +-commutative 99.4%

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

      12. distribute-neg-in 99.4%

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

      13. sub-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. sub-neg 99.4%

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

      16. distribute-neg-in 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -550000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -550000.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -550000.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-550000.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -550000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -550000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -550000.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -550000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -550000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -5.5e5 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. associate-*r/ 99.4%

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

      6. exp-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. distribute-frac-neg2 99.4%

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

      10. distribute-lft-neg-out 99.4%

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

      11. +-commutative 99.4%

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

      12. distribute-neg-in 99.4%

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

      13. sub-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. sub-neg 99.4%

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

      16. distribute-neg-in 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 97.8%

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

Alternative 5: 84.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.0)
   (+ 1.0 (exp b))
   (if (<= b 1.25e+90)
     (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0 + exp(b);
	} else if (b <= 1.25e+90) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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 <= (-2.0d0)) then
        tmp = 1.0d0 + exp(b)
    else if (b <= 1.25d+90) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= -2.0) {
		tmp = 1.0 + Math.exp(b);
	} else if (b <= 1.25e+90) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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 <= -2.0:
		tmp = 1.0 + math.exp(b)
	elif b <= 1.25e+90:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	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 <= -2.0)
		tmp = Float64(1.0 + exp(b));
	elseif (b <= 1.25e+90)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	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 <= -2.0)
		tmp = 1.0 + exp(b);
	elseif (b <= 1.25e+90)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	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, -2.0], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e+90], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 3 regimes

2. if b < -2

   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. *-rgt-identity 100.0%

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

      5. associate-*r/ 100.0%

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

      6. exp-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. distribute-frac-neg2 99.9%

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

      10. distribute-lft-neg-out 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. sub-neg 99.9%

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

      14. *-commutative 99.9%

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

      15. sub-neg 99.9%

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

      16. distribute-neg-in 99.9%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-div 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. div-exp 0.0%

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

      7. log-div 100.0%

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

      8. metadata-eval 100.0%

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

      9. frac-2neg 100.0%

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

      10. div-exp 100.0%

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

      11. log-rec 100.0%

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

      12. log1p-define 100.0%

          \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(\frac{e^{b}}{e^{a}}\right)}} \]

      13. div-exp 100.0%

          \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{e^{b - a}}\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{b - a}\right)}} \]

   7. Taylor expanded in a around 0 100.0%

      \[\leadsto \color{blue}{e^{-\log \left(1 + e^{b}\right)}} \]
   8. Step-by-step derivation

      1. log1p-define 100.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{b}\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 98.1%

          \[\leadsto e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}}} \]

       2. sqrt-unprod 98.5%

          \[\leadsto e^{\color{blue}{\sqrt{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}}} \]

       3. sqr-neg 98.5%

          \[\leadsto e^{\sqrt{\color{blue}{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}}} \]

       4. sqrt-unprod 98.5%

          \[\leadsto e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}}} \]

       5. add-sqr-sqrt 98.5%

          \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

       6. log1p-undefine 98.5%

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

       7. rem-exp-log 98.5%

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

       8. +-commutative 98.5%

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

   11. Applied egg-rr 98.5%

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

   if -2 < b < 1.2500000000000001e90

   1. Initial program 99.3%

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

      1. *-lft-identity 99.3%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.3%

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

      4. *-rgt-identity 99.3%

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

      5. associate-*r/ 99.3%

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

      6. exp-neg 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-neg-frac 99.3%

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

      9. distribute-frac-neg2 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. +-commutative 99.3%

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

      12. distribute-neg-in 99.3%

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

      13. sub-neg 99.3%

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

      14. *-commutative 99.3%

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

      15. sub-neg 99.3%

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

      16. distribute-neg-in 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0 89.6%

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

   6. Taylor expanded in a around 0 79.5%

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

   if 1.2500000000000001e90 < 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. *-rgt-identity 100.0%

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

      5. associate-*r/ 100.0%

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

      6. exp-neg 100.0%

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

      7. metadata-eval 100.0%

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

      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%

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

   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 92.0%

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

      1. *-commutative 92.0%

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

   8. Simplified 92.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 76.9%

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

   7. *-lft-identity 76.9%

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

   8. associate-*l/ 76.9%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   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 7: 70.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.36e+91)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.36e+91) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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 <= 1.36d+91) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= 1.36e+91) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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 <= 1.36e+91:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	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 <= 1.36e+91)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	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 <= 1.36e+91)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	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, 1.36e+91], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < 1.36000000000000007e91

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. associate-*r/ 99.5%

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

      6. exp-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 72.3%

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

   6. Taylor expanded in a around 0 64.1%

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

   if 1.36000000000000007e91 < 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. *-rgt-identity 100.0%

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

      5. associate-*r/ 100.0%

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

      6. exp-neg 100.0%

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

      7. metadata-eval 100.0%

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

      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%

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

   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 92.0%

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

      1. *-commutative 92.0%

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

   8. Simplified 92.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e+149)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e+149) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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) :: tmp
    if (b <= 1.25d+149) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= 1.25e+149) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e+149:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e+149)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	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)
	tmp = 0.0;
	if (b <= 1.25e+149)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e+149], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < 1.24999999999999998e149

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. associate-*r/ 99.5%

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

      6. exp-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 70.2%

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

   6. Taylor expanded in a around 0 62.0%

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

   if 1.24999999999999998e149 < 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. *-rgt-identity 100.0%

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

      5. associate-*r/ 100.0%

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

      6. exp-neg 100.0%

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

      7. metadata-eval 100.0%

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

      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%

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

   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 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9e+148)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9e+148) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} 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) :: tmp
    if (b <= 9d+148) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    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 <= 9e+148) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9e+148:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9e+148)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	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)
	tmp = 0.0;
	if (b <= 9e+148)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9e+148], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $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 < 8.99999999999999987e148

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. associate-*r/ 99.5%

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

      6. exp-neg 99.5%

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

      7. metadata-eval 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 70.2%

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

   6. Taylor expanded in a around 0 59.0%

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

   if 8.99999999999999987e148 < 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. *-rgt-identity 100.0%

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

      5. associate-*r/ 100.0%

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

      6. exp-neg 100.0%

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

      7. metadata-eval 100.0%

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

      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%

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

   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 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.6%

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

   6. exp-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in b around 0 65.4%

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

6. Taylor expanded in a around 0 53.9%

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

7. Final simplification 53.9%

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

Alternative 11: 40.3% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.6%

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

   6. exp-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in b around 0 65.4%

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

6. Taylor expanded in a around 0 42.7%

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

   1. neg-mul-1 42.7%

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

   2. unsub-neg 42.7%

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

8. Simplified 42.7%

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

Alternative 12: 39.6% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.6%

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

   6. exp-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in b around 0 65.4%

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

6. Taylor expanded in a around 0 42.6%

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

   1. *-commutative 42.6%

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

8. Simplified 42.6%

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

Alternative 13: 39.5% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.6%

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

   6. exp-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 83.2%

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

6. Taylor expanded in b around 0 41.5%

   \[\leadsto \color{blue}{0.5} \]
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 2024151 
(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))))