Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 11.2s

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-/l* 98.0%

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

   3. remove-double-div 98.0%

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

   4. exp-neg 98.0%

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

   5. associate-/r/ 98.0%

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

   6. /-rgt-identity 98.0%

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

   7. *-commutative 98.0%

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

   8. distribute-rgt-in 69.9%

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

   9. exp-neg 69.9%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.022 \lor \neg \left(a \leq 0.00018\right):\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.022) (not (<= a 0.00018)))
   (/ 1.0 (+ 1.0 (exp (- a))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.022) || !(a <= 0.00018)) {
		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 <= (-0.022d0)) .or. (.not. (a <= 0.00018d0))) 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 <= -0.022) || !(a <= 0.00018)) {
		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 <= -0.022) or not (a <= 0.00018):
		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 <= -0.022) || !(a <= 0.00018))
		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 <= -0.022) || ~((a <= 0.00018)))
		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[Or[LessEqual[a, -0.022], N[Not[LessEqual[a, 0.00018]], $MachinePrecision]], 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 < -0.021999999999999999 or 1.80000000000000011e-4 < a

   1. Initial program 93.7%

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

      1. *-lft-identity 93.7%

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

      2. associate-/l* 93.7%

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

      3. remove-double-div 93.7%

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

      4. exp-neg 93.7%

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

      5. associate-/r/ 93.7%

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

      6. /-rgt-identity 93.7%

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

      7. *-commutative 93.7%

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

      8. distribute-rgt-in 2.5%

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

      9. exp-neg 2.5%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   if -0.021999999999999999 < a < 1.80000000000000011e-4

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.022 \lor \neg \left(a \leq 0.00018\right):\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 92.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{a \cdot a + a \cdot 2}{\left(-a\right) \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e+154)
   (/ 2.0 (* a a))
   (if (<= a -1.6e+102)
     (/ (+ (* a a) (* a 2.0)) (* (- a) (* a a)))
     (/ 1.0 (+ 1.0 (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+154) {
		tmp = 2.0 / (a * a);
	} else if (a <= -1.6e+102) {
		tmp = ((a * a) + (a * 2.0)) / (-a * (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 <= (-1.35d+154)) then
        tmp = 2.0d0 / (a * a)
    else if (a <= (-1.6d+102)) then
        tmp = ((a * a) + (a * 2.0d0)) / (-a * (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 <= -1.35e+154) {
		tmp = 2.0 / (a * a);
	} else if (a <= -1.6e+102) {
		tmp = ((a * a) + (a * 2.0)) / (-a * (a * a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35e+154:
		tmp = 2.0 / (a * a)
	elif a <= -1.6e+102:
		tmp = ((a * a) + (a * 2.0)) / (-a * (a * a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35e+154)
		tmp = Float64(2.0 / Float64(a * a));
	elseif (a <= -1.6e+102)
		tmp = Float64(Float64(Float64(a * a) + Float64(a * 2.0)) / Float64(Float64(-a) * Float64(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 <= -1.35e+154)
		tmp = 2.0 / (a * a);
	elseif (a <= -1.6e+102)
		tmp = ((a * a) + (a * 2.0)) / (-a * (a * a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35e+154], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.6e+102], N[(N[(N[(a * a), $MachinePrecision] + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[((-a) * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.35000000000000003e154

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around inf 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \frac{2}{\color{blue}{a \cdot a}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{a \cdot a}} \]

   if -1.35000000000000003e154 < a < -1.6e102

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 4.3%

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

      1. neg-mul-1 4.3%

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

      2. unsub-neg 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in a around inf 4.3%

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

      1. distribute-neg-in 4.3%

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

      2. unsub-neg 4.3%

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

      3. distribute-neg-frac 4.3%

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

      4. metadata-eval 4.3%

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

      5. associate-*r/ 4.3%

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

      6. metadata-eval 4.3%

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

      7. unpow2 4.3%

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

      8. associate-/r* 4.3%

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

   10. Simplified 4.3%

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

       1. frac-2neg 4.3%

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

       2. metadata-eval 4.3%

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

       3. associate-/l/ 4.3%

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

       4. frac-sub 89.3%

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

       5. *-un-lft-identity 89.3%

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

   12. Applied egg-rr 89.3%

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

   if -1.6e102 < a

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. associate-/l* 97.6%

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

      3. remove-double-div 97.5%

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

      4. exp-neg 97.5%

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

      5. associate-/r/ 97.6%

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

      6. /-rgt-identity 97.6%

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

      7. *-commutative 97.6%

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

      8. distribute-rgt-in 86.4%

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

      9. exp-neg 86.4%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 90.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{a \cdot a + a \cdot 2}{\left(-a\right) \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 82.4% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -2.7:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0 - b}{b \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (* b b))))
   (if (<= b -2.7)
     1.0
     (if (<= b 7.6e+62)
       (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
       (if (<= b 1.35e+154) (/ (- t_0 b) (* b t_0)) (/ -2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= -2.7) {
		tmp = 1.0;
	} else if (b <= 7.6e+62) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	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 = 0.5d0 * (b * b)
    if (b <= (-2.7d0)) then
        tmp = 1.0d0
    else if (b <= 7.6d+62) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else if (b <= 1.35d+154) then
        tmp = (t_0 - b) / (b * t_0)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= -2.7) {
		tmp = 1.0;
	} else if (b <= 7.6e+62) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * (b * b)
	tmp = 0
	if b <= -2.7:
		tmp = 1.0
	elif b <= 7.6e+62:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	elif b <= 1.35e+154:
		tmp = (t_0 - b) / (b * t_0)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(b * b))
	tmp = 0.0
	if (b <= -2.7)
		tmp = 1.0;
	elseif (b <= 7.6e+62)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(t_0 - b) / Float64(b * t_0));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * (b * b);
	tmp = 0.0;
	if (b <= -2.7)
		tmp = 1.0;
	elseif (b <= 7.6e+62)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	elseif (b <= 1.35e+154)
		tmp = (t_0 - b) / (b * t_0);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.7], 1.0, If[LessEqual[b, 7.6e+62], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(t$95$0 - b), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.7000000000000002

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -2.7000000000000002 < b < 7.59999999999999967e62

   1. Initial program 97.4%

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

      1. *-lft-identity 97.4%

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

      2. associate-/l* 97.4%

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

      3. remove-double-div 97.4%

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

      4. exp-neg 97.4%

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

      5. associate-/r/ 97.4%

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

      6. /-rgt-identity 97.4%

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

      7. *-commutative 97.4%

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

      8. distribute-rgt-in 62.8%

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

      9. exp-neg 62.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 95.3%

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

   5. Taylor expanded in a around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. neg-mul-1 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. associate-*r* 77.9%

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

   7. Simplified 77.9%

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

      1. *-un-lft-identity 77.9%

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

      2. distribute-rgt-out-- 77.9%

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

   9. Applied egg-rr 77.9%

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

   if 7.59999999999999967e62 < b < 1.35000000000000003e154

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 76.0%

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

      9. exp-neg 76.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 4.2%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 4.2%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 4.2%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 4.2%

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

      1. associate-*r/ 4.2%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 4.2%

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

      3. unpow2 4.2%

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

   10. Simplified 4.2%

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

       1. clear-num 4.2%

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

       2. frac-sub 69.2%

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

       3. *-un-lft-identity 69.2%

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

       4. div-inv 69.2%

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

       5. metadata-eval 69.2%

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

       6. *-commutative 69.2%

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

       7. *-un-lft-identity 69.2%

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

       8. div-inv 69.2%

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

       9. metadata-eval 69.2%

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

   12. Applied egg-rr 69.2%

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

   if 1.35000000000000003e154 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 60.0%

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

      9. exp-neg 60.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 7.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 7.3%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 7.3%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 7.3%

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

      1. associate-*r/ 7.3%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 7.3%

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

      3. unpow2 7.3%

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

   10. Simplified 7.3%

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

   11. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(b \cdot b\right) - b}{b \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 5: 79.4% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.4)
   1.0
   (if (<= b 2.6e+141)
     (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
     (/ -2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.4) {
		tmp = 1.0;
	} else if (b <= 2.6e+141) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.4d0)) then
        tmp = 1.0d0
    else if (b <= 2.6d+141) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.4) {
		tmp = 1.0;
	} else if (b <= 2.6e+141) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.4:
		tmp = 1.0
	elif b <= 2.6e+141:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.4)
		tmp = 1.0;
	elseif (b <= 2.6e+141)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.4)
		tmp = 1.0;
	elseif (b <= 2.6e+141)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.4], 1.0, If[LessEqual[b, 2.6e+141], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4000000000000004

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -4.4000000000000004 < b < 2.5999999999999999e141

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 63.6%

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

      9. exp-neg 63.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 88.2%

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

   5. Taylor expanded in a around 0 72.1%

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

      1. +-commutative 72.1%

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

      2. neg-mul-1 72.1%

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

      3. unsub-neg 72.1%

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

      4. unpow2 72.1%

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

      5. associate-*r* 72.1%

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

   7. Simplified 72.1%

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

      1. *-un-lft-identity 72.1%

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

      2. distribute-rgt-out-- 72.1%

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

   9. Applied egg-rr 72.1%

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

   if 2.5999999999999999e141 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.7%

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

      9. exp-neg 65.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 6.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 6.9%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 6.9%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 6.9%

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

      1. associate-*r/ 6.9%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 6.9%

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

      3. unpow2 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in b around 0 87.4%

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

       1. unpow2 87.4%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 87.4%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 6: 79.4% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(b \cdot b\right) + \left(b + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.5)
   1.0
   (if (<= b 2.65e+141)
     (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
     (/ 1.0 (+ (* 0.5 (* b b)) (+ b 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -5.5) {
		tmp = 1.0;
	} else if (b <= 2.65e+141) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / ((0.5 * (b * b)) + (b + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-5.5d0)) then
        tmp = 1.0d0
    else if (b <= 2.65d+141) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / ((0.5d0 * (b * b)) + (b + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -5.5) {
		tmp = 1.0;
	} else if (b <= 2.65e+141) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / ((0.5 * (b * b)) + (b + 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -5.5:
		tmp = 1.0
	elif b <= 2.65e+141:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / ((0.5 * (b * b)) + (b + 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -5.5)
		tmp = 1.0;
	elseif (b <= 2.65e+141)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(b * b)) + Float64(b + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -5.5)
		tmp = 1.0;
	elseif (b <= 2.65e+141)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / ((0.5 * (b * b)) + (b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -5.5], 1.0, If[LessEqual[b, 2.65e+141], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -5.5 < b < 2.65e141

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 63.6%

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

      9. exp-neg 63.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 88.2%

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

   5. Taylor expanded in a around 0 72.1%

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

      1. +-commutative 72.1%

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

      2. neg-mul-1 72.1%

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

      3. unsub-neg 72.1%

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

      4. unpow2 72.1%

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

      5. associate-*r* 72.1%

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

   7. Simplified 72.1%

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

      1. *-un-lft-identity 72.1%

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

      2. distribute-rgt-out-- 72.1%

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

   9. Applied egg-rr 72.1%

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

   if 2.65e141 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.7%

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

      9. exp-neg 65.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 87.4%

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

      1. associate-+r+ 87.4%

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

      2. +-commutative 87.4%

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

      3. *-commutative 87.4%

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

      4. unpow2 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(b \cdot b\right) + \left(b + 2\right)}\\ \end{array} \]

Alternative 7: 79.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -10.5)
   1.0
   (if (<= b 2.35e+141) (/ (+ a 2.0) (- 4.0 (* a a))) (/ -2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -10.5) {
		tmp = 1.0;
	} else if (b <= 2.35e+141) {
		tmp = (a + 2.0) / (4.0 - (a * a));
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-10.5d0)) then
        tmp = 1.0d0
    else if (b <= 2.35d+141) then
        tmp = (a + 2.0d0) / (4.0d0 - (a * a))
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -10.5) {
		tmp = 1.0;
	} else if (b <= 2.35e+141) {
		tmp = (a + 2.0) / (4.0 - (a * a));
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -10.5:
		tmp = 1.0
	elif b <= 2.35e+141:
		tmp = (a + 2.0) / (4.0 - (a * a))
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -10.5)
		tmp = 1.0;
	elseif (b <= 2.35e+141)
		tmp = Float64(Float64(a + 2.0) / Float64(4.0 - Float64(a * a)));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -10.5)
		tmp = 1.0;
	elseif (b <= 2.35e+141)
		tmp = (a + 2.0) / (4.0 - (a * a));
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -10.5], 1.0, If[LessEqual[b, 2.35e+141], N[(N[(a + 2.0), $MachinePrecision] / N[(4.0 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -10.5

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -10.5 < b < 2.3499999999999999e141

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 63.6%

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

      9. exp-neg 63.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 88.2%

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

   5. Taylor expanded in a around 0 53.1%

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

      1. neg-mul-1 53.1%

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

      2. unsub-neg 53.1%

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

   7. Simplified 53.1%

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

      1. flip-- 71.6%

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

      2. associate-/r/ 71.6%

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

      3. metadata-eval 71.6%

         \[\leadsto \frac{1}{\color{blue}{4} - a \cdot a} \cdot \left(2 + a\right) \]

   9. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{1}{4 - a \cdot a} \cdot \left(2 + a\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 71.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(2 + a\right)}{4 - a \cdot a}} \]

       2. *-lft-identity 71.6%

          \[\leadsto \frac{\color{blue}{2 + a}}{4 - a \cdot a} \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{2 + a}{4 - a \cdot a}} \]

   if 2.3499999999999999e141 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.7%

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

      9. exp-neg 65.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 6.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 6.9%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 6.9%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 6.9%

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

      1. associate-*r/ 6.9%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 6.9%

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

      3. unpow2 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in b around 0 87.4%

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

       1. unpow2 87.4%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 87.4%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -10.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 8: 70.6% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.175:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 185:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.175)
   1.0
   (if (<= b 185.0)
     (/ 1.0 (- 2.0 a))
     (if (<= b 2.9e+139) (/ 2.0 (* a a)) (/ -2.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.175) {
		tmp = 1.0;
	} else if (b <= 185.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 2.9e+139) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.175d0)) then
        tmp = 1.0d0
    else if (b <= 185.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 2.9d+139) then
        tmp = 2.0d0 / (a * a)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.175) {
		tmp = 1.0;
	} else if (b <= 185.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 2.9e+139) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.175:
		tmp = 1.0
	elif b <= 185.0:
		tmp = 1.0 / (2.0 - a)
	elif b <= 2.9e+139:
		tmp = 2.0 / (a * a)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.175)
		tmp = 1.0;
	elseif (b <= 185.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 2.9e+139)
		tmp = Float64(2.0 / Float64(a * a));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.175)
		tmp = 1.0;
	elseif (b <= 185.0)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 2.9e+139)
		tmp = 2.0 / (a * a);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.175], 1.0, If[LessEqual[b, 185.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e+139], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.17499999999999999

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -0.17499999999999999 < b < 185

   1. Initial program 97.2%

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

      1. *-lft-identity 97.2%

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

      2. associate-/l* 97.2%

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

      3. remove-double-div 97.2%

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

      4. exp-neg 97.2%

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

      5. associate-/r/ 97.2%

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

      6. /-rgt-identity 97.2%

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

      7. *-commutative 97.2%

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

      8. distribute-rgt-in 63.9%

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

      9. exp-neg 63.9%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 97.7%

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

   5. Taylor expanded in a around 0 62.8%

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

      1. neg-mul-1 62.8%

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

      2. unsub-neg 62.8%

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

   7. Simplified 62.8%

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

   if 185 < b < 2.8999999999999999e139

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 62.1%

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

      9. exp-neg 62.1%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 39.9%

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

   5. Taylor expanded in a around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. neg-mul-1 30.0%

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

      3. unsub-neg 30.0%

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

      4. unpow2 30.0%

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

      5. associate-*r* 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in a around inf 29.2%

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

      1. unpow2 29.2%

         \[\leadsto \frac{2}{\color{blue}{a \cdot a}} \]

   10. Simplified 29.2%

       \[\leadsto \color{blue}{\frac{2}{a \cdot a}} \]

   if 2.8999999999999999e139 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.7%

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

      9. exp-neg 65.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 6.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 6.9%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 6.9%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 6.9%

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

      1. associate-*r/ 6.9%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 6.9%

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

      3. unpow2 6.9%

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

   10. Simplified 6.9%

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

   11. Taylor expanded in b around 0 87.4%

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

       1. unpow2 87.4%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 87.4%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.175:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 185:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 9: 68.5% accurate, 33.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.55) 1.0 (if (<= b 2.9) (+ 0.5 (* b -0.25)) (/ -2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.55) {
		tmp = 1.0;
	} else if (b <= 2.9) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.55d0)) then
        tmp = 1.0d0
    else if (b <= 2.9d0) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.55) {
		tmp = 1.0;
	} else if (b <= 2.9) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.55:
		tmp = 1.0
	elif b <= 2.9:
		tmp = 0.5 + (b * -0.25)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.55)
		tmp = 1.0;
	elseif (b <= 2.9)
		tmp = Float64(0.5 + Float64(b * -0.25));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.55)
		tmp = 1.0;
	elseif (b <= 2.9)
		tmp = 0.5 + (b * -0.25);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.55], 1.0, If[LessEqual[b, 2.9], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.55000000000000004

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.55000000000000004 < b < 2.89999999999999991

   1. Initial program 97.2%

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

      1. *-lft-identity 97.2%

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

      2. associate-/l* 97.2%

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

      3. remove-double-div 97.2%

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

      4. exp-neg 97.2%

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

      5. associate-/r/ 97.2%

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

      6. /-rgt-identity 97.2%

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

      7. *-commutative 97.2%

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

      8. distribute-rgt-in 63.4%

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

      9. exp-neg 63.4%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 64.0%

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

   5. Taylor expanded in b around 0 63.6%

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

      1. *-commutative 63.6%

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

   7. Simplified 63.6%

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

   if 2.89999999999999991 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.2%

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

      9. exp-neg 65.2%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 5.7%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 5.7%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 5.7%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 5.7%

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

      1. associate-*r/ 5.7%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 5.7%

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

      3. unpow2 5.7%

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

   10. Simplified 5.7%

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

   11. Taylor expanded in b around 0 48.7%

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

       1. unpow2 48.7%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 48.7%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 10: 68.7% accurate, 33.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.54:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.54) 1.0 (if (<= b 6.2e+54) (/ 1.0 (- 2.0 a)) (/ -2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.54) {
		tmp = 1.0;
	} else if (b <= 6.2e+54) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.54d0)) then
        tmp = 1.0d0
    else if (b <= 6.2d+54) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.54) {
		tmp = 1.0;
	} else if (b <= 6.2e+54) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.54:
		tmp = 1.0
	elif b <= 6.2e+54:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.54)
		tmp = 1.0;
	elseif (b <= 6.2e+54)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.54)
		tmp = 1.0;
	elseif (b <= 6.2e+54)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.54], 1.0, If[LessEqual[b, 6.2e+54], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.54000000000000004

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -0.54000000000000004 < b < 6.1999999999999999e54

   1. Initial program 97.4%

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

      1. *-lft-identity 97.4%

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

      2. associate-/l* 97.4%

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

      3. remove-double-div 97.4%

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

      4. exp-neg 97.4%

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

      5. associate-/r/ 97.4%

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

      6. /-rgt-identity 97.4%

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

      7. *-commutative 97.4%

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

      8. distribute-rgt-in 62.8%

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

      9. exp-neg 62.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 95.3%

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

   5. Taylor expanded in a around 0 59.4%

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

      1. neg-mul-1 59.4%

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

      2. unsub-neg 59.4%

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

   7. Simplified 59.4%

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

   if 6.1999999999999999e54 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 67.3%

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

      9. exp-neg 67.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 5.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   6. Step-by-step derivation

      1. +-commutative 5.9%

         \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   7. Simplified 5.9%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]

   8. Taylor expanded in b around inf 5.9%

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

      1. associate-*r/ 5.9%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 5.9%

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

      3. unpow2 5.9%

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

   10. Simplified 5.9%

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

   11. Taylor expanded in b around 0 57.8%

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

       1. unpow2 57.8%

          \[\leadsto \frac{-2}{\color{blue}{b \cdot b}} \]

   13. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{-2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.54:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 11: 54.2% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.25) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.25d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.25:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.25)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.25], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.25

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.25 < b

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-/l* 98.1%

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

      3. remove-double-div 98.1%

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

      4. exp-neg 98.1%

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

      5. associate-/r/ 98.1%

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

      6. /-rgt-identity 98.1%

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

      7. *-commutative 98.1%

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

      8. distribute-rgt-in 64.0%

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

      9. exp-neg 64.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 75.2%

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

   5. Taylor expanded in b around 0 44.4%

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

      1. *-commutative 44.4%

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

   7. Simplified 44.4%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]

Alternative 12: 53.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.9) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.9) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-0.9d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.9) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.9:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.9)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.9)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.9], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -0.900000000000000022

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. exp-neg 100.0%

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

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]

   if -0.900000000000000022 < b

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-/l* 98.1%

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

      3. remove-double-div 98.1%

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

      4. exp-neg 98.1%

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

      5. associate-/r/ 98.1%

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

      6. /-rgt-identity 98.1%

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

      7. *-commutative 98.1%

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

      8. distribute-rgt-in 64.0%

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

      9. exp-neg 64.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 75.2%

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

   5. Taylor expanded in b around 0 44.3%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-/l* 98.0%

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

   3. remove-double-div 98.0%

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

   4. exp-neg 98.0%

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

   5. associate-/r/ 98.0%

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

   6. /-rgt-identity 98.0%

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

   7. *-commutative 98.0%

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

   8. distribute-rgt-in 69.9%

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

   9. exp-neg 69.9%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 79.6%

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

5. Taylor expanded in b around 0 39.8%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 39.8%

   \[\leadsto 0.5 \]

Developer target: 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 2023285 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))