Quotient of sum of exps

Percentage Accurate: 98.6% → 100.0%

Time: 6.9s

Alternatives: 11

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: 98.6% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 73.0%

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

   9. exp-neg 73.0%

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

   10. rgt-mult-inverse 99.6%

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

   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: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -490000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -490000000.0) (/ (exp a) b) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -490000000.0) {
		tmp = exp(a) / b;
	} 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 <= (-490000000.0d0)) then
        tmp = exp(a) / b
    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 <= -490000000.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -490000000.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -490000000.0)
		tmp = Float64(exp(a) / b);
	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 <= -490000000.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -490000000.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9e8

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in b around inf 100.0%

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

   if -4.9e8 < a

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-/l* 98.4%

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 97.4%

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

      9. exp-neg 97.4%

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

      10. rgt-mult-inverse 99.5%

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

      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 97.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -490000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 71.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1200:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1200.0) (/ (exp a) b) (/ 1.0 (+ 2.0 (+ b (* (* b b) 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1200.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1200.0d0)) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (2.0d0 + (b + ((b * b) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1200.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1200.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1200.0)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(Float64(b * b) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1200.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1200.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1200

   1. Initial program 98.5%

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

   2. Taylor expanded in b around 0 98.5%

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

   3. Taylor expanded in b around inf 98.5%

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

   if -1200 < a

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.4%

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

      9. exp-neg 98.4%

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

      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 97.4%

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

   5. Taylor expanded in b around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

   7. Simplified 62.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1200:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 4: 68.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot -0.5\right)\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{t_0 \cdot t_0 - b \cdot b}{\left(-b\right) - t_0} - -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b -0.5))))
   (if (<= b 2.5e+75)
     (/ (+ a 2.0) (- 4.0 (* a a)))
     (if (<= b 1.35e+154)
       (/ 1.0 (- (/ (- (* t_0 t_0) (* b b)) (- (- b) t_0)) -2.0))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = b * (b * -0.5);
	double tmp;
	if (b <= 2.5e+75) {
		tmp = (a + 2.0) / (4.0 - (a * a));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((t_0 * t_0) - (b * b)) / (-b - t_0)) - -2.0);
	} 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 = b * (b * (-0.5d0))
    if (b <= 2.5d+75) then
        tmp = (a + 2.0d0) / (4.0d0 - (a * a))
    else if (b <= 1.35d+154) then
        tmp = 1.0d0 / ((((t_0 * t_0) - (b * b)) / (-b - t_0)) - (-2.0d0))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * -0.5);
	double tmp;
	if (b <= 2.5e+75) {
		tmp = (a + 2.0) / (4.0 - (a * a));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((t_0 * t_0) - (b * b)) / (-b - t_0)) - -2.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * -0.5)
	tmp = 0
	if b <= 2.5e+75:
		tmp = (a + 2.0) / (4.0 - (a * a))
	elif b <= 1.35e+154:
		tmp = 1.0 / ((((t_0 * t_0) - (b * b)) / (-b - t_0)) - -2.0)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * -0.5))
	tmp = 0.0
	if (b <= 2.5e+75)
		tmp = Float64(Float64(a + 2.0) / Float64(4.0 - Float64(a * a)));
	elseif (b <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(b * b)) / Float64(Float64(-b) - t_0)) - -2.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * -0.5);
	tmp = 0.0;
	if (b <= 2.5e+75)
		tmp = (a + 2.0) / (4.0 - (a * a));
	elseif (b <= 1.35e+154)
		tmp = 1.0 / ((((t_0 * t_0) - (b * b)) / (-b - t_0)) - -2.0);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.5e+75], N[(N[(a + 2.0), $MachinePrecision] / N[(4.0 - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(1.0 / N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.5000000000000001e75

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-/l* 98.5%

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

      3. remove-double-div 98.5%

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

      4. exp-neg 98.5%

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

      5. associate-/r/ 98.5%

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

      6. /-rgt-identity 98.5%

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

      7. *-commutative 98.5%

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

      8. distribute-rgt-in 74.7%

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

      9. exp-neg 74.7%

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

      10. rgt-mult-inverse 99.5%

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

      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 74.6%

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

   5. Taylor expanded in a around 0 49.9%

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

      1. neg-mul-1 49.9%

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

      2. unsub-neg 49.9%

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

   7. Simplified 49.9%

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

      1. flip-- 64.2%

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

      2. associate-/r/ 64.2%

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

      3. metadata-eval 64.2%

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

   9. Applied egg-rr 64.2%

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

       1. associate-*l/ 64.2%

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

       2. *-lft-identity 64.2%

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

   11. Simplified 64.2%

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

   if 2.5000000000000001e75 < 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 63.2%

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

      9. exp-neg 63.2%

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

      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%

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

   5. Taylor expanded in b around 0 7.1%

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

      1. *-commutative 7.1%

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

      2. unpow2 7.1%

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

   7. Simplified 7.1%

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

      1. /-rgt-identity 7.1%

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

      2. frac-2neg 7.1%

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

      3. +-commutative 7.1%

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

      4. associate-+l+ 7.1%

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

      5. associate-*l* 7.1%

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

      6. fma-def 7.1%

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

      7. metadata-eval 7.1%

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

   9. Applied egg-rr 7.1%

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

       1. distribute-frac-neg 7.1%

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

       2. *-lft-identity 7.1%

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

       3. associate-/l* 7.1%

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

       4. metadata-eval 7.1%

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

       5. associate-/r* 7.1%

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

       6. neg-mul-1 7.1%

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

       7. remove-double-div 7.1%

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

       8. +-commutative 7.1%

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

       9. fma-udef 7.1%

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

       10. +-commutative 7.1%

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

       11. associate-+r+ 7.1%

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

       12. fma-udef 7.1%

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

       13. distribute-neg-in 7.1%

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

       14. metadata-eval 7.1%

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

       15. fma-udef 7.1%

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

       16. +-commutative 7.1%

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

       17. distribute-neg-in 7.1%

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

       18. neg-mul-1 7.1%

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

       19. *-commutative 7.1%

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

       20. distribute-rgt-neg-in 7.1%

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

       21. *-commutative 7.1%

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

   11. Simplified 7.1%

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

       1. distribute-rgt-in 7.1%

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

       2. flip-+ 100.0%

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

       3. mul-1-neg 100.0%

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

       4. mul-1-neg 100.0%

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

       5. associate-*r* 100.0%

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

       6. *-commutative 100.0%

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

       7. associate-*r* 100.0%

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

       8. *-commutative 100.0%

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

       9. mul-1-neg 100.0%

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

       10. associate-*r* 100.0%

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

       11. *-commutative 100.0%

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

   13. Applied egg-rr 100.0%

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

   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 68.6%

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

      9. exp-neg 68.6%

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

      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 a around 0 100.0%

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

   5. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\left(b \cdot \left(b \cdot -0.5\right)\right) \cdot \left(b \cdot \left(b \cdot -0.5\right)\right) - b \cdot b}{\left(-b\right) - b \cdot \left(b \cdot -0.5\right)} - -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 5: 65.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{-1}{a \cdot a}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{1}{\frac{16 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{4 + a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+154)
   (* (+ a 2.0) (/ -1.0 (* a a)))
   (if (<= a -2.5e+58)
     (* (+ a 2.0) (/ 1.0 (/ (- 16.0 (* (* a a) (* a a))) (+ 4.0 (* a a)))))
     (/ 1.0 (+ 2.0 (+ b (* (* b b) 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+154) {
		tmp = (a + 2.0) * (-1.0 / (a * a));
	} else if (a <= -2.5e+58) {
		tmp = (a + 2.0) * (1.0 / ((16.0 - ((a * a) * (a * a))) / (4.0 + (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.4d+154)) then
        tmp = (a + 2.0d0) * ((-1.0d0) / (a * a))
    else if (a <= (-2.5d+58)) then
        tmp = (a + 2.0d0) * (1.0d0 / ((16.0d0 - ((a * a) * (a * a))) / (4.0d0 + (a * a))))
    else
        tmp = 1.0d0 / (2.0d0 + (b + ((b * b) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+154) {
		tmp = (a + 2.0) * (-1.0 / (a * a));
	} else if (a <= -2.5e+58) {
		tmp = (a + 2.0) * (1.0 / ((16.0 - ((a * a) * (a * a))) / (4.0 + (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e+154:
		tmp = (a + 2.0) * (-1.0 / (a * a))
	elif a <= -2.5e+58:
		tmp = (a + 2.0) * (1.0 / ((16.0 - ((a * a) * (a * a))) / (4.0 + (a * a))))
	else:
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e+154)
		tmp = Float64(Float64(a + 2.0) * Float64(-1.0 / Float64(a * a)));
	elseif (a <= -2.5e+58)
		tmp = Float64(Float64(a + 2.0) * Float64(1.0 / Float64(Float64(16.0 - Float64(Float64(a * a) * Float64(a * a))) / Float64(4.0 + Float64(a * a)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(Float64(b * b) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e+154)
		tmp = (a + 2.0) * (-1.0 / (a * a));
	elseif (a <= -2.5e+58)
		tmp = (a + 2.0) * (1.0 / ((16.0 - ((a * a) * (a * a))) / (4.0 + (a * a))));
	else
		tmp = 1.0 / (2.0 + (b + ((b * b) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e+154], N[(N[(a + 2.0), $MachinePrecision] * N[(-1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.5e+58], N[(N[(a + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(16.0 - N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.4e154

   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%

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

   5. Taylor expanded in a around 0 6.7%

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

      1. neg-mul-1 6.7%

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

      2. unsub-neg 6.7%

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

   7. Simplified 6.7%

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

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in a around inf 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

   if -1.4e154 < a < -2.49999999999999993e58

   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%

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

   5. Taylor expanded in a around 0 4.0%

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

      1. neg-mul-1 4.0%

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

      2. unsub-neg 4.0%

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

   7. Simplified 4.0%

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

      1. flip-- 4.0%

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

      2. associate-/r/ 4.0%

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

      3. metadata-eval 4.0%

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

   9. Applied egg-rr 4.0%

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

       1. sub-neg 4.0%

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

       2. flip-+ 85.2%

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

       3. metadata-eval 85.2%

          \[\leadsto \frac{1}{\frac{\color{blue}{16} - \left(-a \cdot a\right) \cdot \left(-a \cdot a\right)}{4 - \left(-a \cdot a\right)}} \cdot \left(2 + a\right) \]

       4. distribute-rgt-neg-in 85.2%

          \[\leadsto \frac{1}{\frac{16 - \color{blue}{\left(a \cdot \left(-a\right)\right)} \cdot \left(-a \cdot a\right)}{4 - \left(-a \cdot a\right)}} \cdot \left(2 + a\right) \]

       5. distribute-rgt-neg-in 85.2%

          \[\leadsto \frac{1}{\frac{16 - \left(a \cdot \left(-a\right)\right) \cdot \color{blue}{\left(a \cdot \left(-a\right)\right)}}{4 - \left(-a \cdot a\right)}} \cdot \left(2 + a\right) \]

       6. distribute-rgt-neg-in 85.2%

          \[\leadsto \frac{1}{\frac{16 - \left(a \cdot \left(-a\right)\right) \cdot \left(a \cdot \left(-a\right)\right)}{4 - \color{blue}{a \cdot \left(-a\right)}}} \cdot \left(2 + a\right) \]

   11. Applied egg-rr 85.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{16 - \left(a \cdot \left(-a\right)\right) \cdot \left(a \cdot \left(-a\right)\right)}{4 - a \cdot \left(-a\right)}}} \cdot \left(2 + a\right) \]

   if -2.49999999999999993e58 < a

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-/l* 98.5%

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

      3. remove-double-div 98.5%

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

      4. exp-neg 98.5%

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

      5. associate-/r/ 98.5%

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

      6. /-rgt-identity 98.5%

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

      7. *-commutative 98.5%

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

      8. distribute-rgt-in 92.1%

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

      9. exp-neg 92.1%

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

      10. rgt-mult-inverse 99.5%

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

      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 94.7%

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

   5. Taylor expanded in b around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. unpow2 61.1%

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

   7. Simplified 61.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{-1}{a \cdot a}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{1}{\frac{16 - \left(a \cdot a\right) \cdot \left(a \cdot a\right)}{4 + a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + \left(b \cdot b\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 6: 53.2% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{+64}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{-1}{a \cdot a}\\ \mathbf{elif}\;a \leq -0.75:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.4e+64)
   (* (+ a 2.0) (/ -1.0 (* a a)))
   (if (<= a -0.75) (/ 2.0 (* b b)) (+ 0.5 (* a 0.25)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.4e+64) {
		tmp = (a + 2.0) * (-1.0 / (a * a));
	} else if (a <= -0.75) {
		tmp = 2.0 / (b * b);
	} else {
		tmp = 0.5 + (a * 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 (a <= (-9.4d+64)) then
        tmp = (a + 2.0d0) * ((-1.0d0) / (a * a))
    else if (a <= (-0.75d0)) then
        tmp = 2.0d0 / (b * b)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.4e+64) {
		tmp = (a + 2.0) * (-1.0 / (a * a));
	} else if (a <= -0.75) {
		tmp = 2.0 / (b * b);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.4e+64:
		tmp = (a + 2.0) * (-1.0 / (a * a))
	elif a <= -0.75:
		tmp = 2.0 / (b * b)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.4e+64)
		tmp = Float64(Float64(a + 2.0) * Float64(-1.0 / Float64(a * a)));
	elseif (a <= -0.75)
		tmp = Float64(2.0 / Float64(b * b));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.4e+64)
		tmp = (a + 2.0) * (-1.0 / (a * a));
	elseif (a <= -0.75)
		tmp = 2.0 / (b * b);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.4e+64], N[(N[(a + 2.0), $MachinePrecision] * N[(-1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -0.75], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.40000000000000058e64

   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%

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

   5. Taylor expanded in a around 0 6.0%

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

      1. neg-mul-1 6.0%

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

      2. unsub-neg 6.0%

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

   7. Simplified 6.0%

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

      1. flip-- 76.4%

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

      2. associate-/r/ 76.4%

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

      3. metadata-eval 76.4%

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

   9. Applied egg-rr 76.4%

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

   10. Taylor expanded in a around inf 76.4%

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

       1. unpow2 76.4%

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

   12. Simplified 76.4%

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

   if -9.40000000000000058e64 < a < -0.75

   1. Initial program 93.3%

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

      1. *-lft-identity 93.3%

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

      2. associate-/l* 93.3%

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

      3. remove-double-div 93.3%

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

      4. exp-neg 93.3%

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

      5. associate-/r/ 93.3%

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

      6. /-rgt-identity 93.3%

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

      7. *-commutative 93.3%

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

      8. distribute-rgt-in 6.7%

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

      9. exp-neg 6.7%

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

      10. rgt-mult-inverse 93.3%

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

      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 61.3%

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

   5. Taylor expanded in b around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. unpow2 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in b around inf 48.1%

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

      1. unpow2 48.1%

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

   10. Simplified 48.1%

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

   if -0.75 < a

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      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 55.2%

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

   5. Taylor expanded in a around 0 53.2%

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

      1. *-commutative 53.2%

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

   7. Simplified 53.2%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{+64}:\\ \;\;\;\;\left(a + 2\right) \cdot \frac{-1}{a \cdot a}\\ \mathbf{elif}\;a \leq -0.75:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 7: 63.0% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e+152) (/ (+ a 2.0) (- 4.0 (* a a))) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e+152) {
		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 <= 3.7d+152) 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 <= 3.7e+152) {
		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 <= 3.7e+152:
		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 <= 3.7e+152)
		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 <= 3.7e+152)
		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, 3.7e+152], 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 2 regimes

2. if b < 3.69999999999999996e152

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 73.6%

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

      9. exp-neg 73.6%

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

      10. rgt-mult-inverse 99.5%

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

      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 71.9%

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

   5. Taylor expanded in a around 0 46.1%

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

      1. neg-mul-1 46.1%

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

      2. unsub-neg 46.1%

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

   7. Simplified 46.1%

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

      1. flip-- 61.0%

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

      2. associate-/r/ 61.0%

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

      3. metadata-eval 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. associate-*l/ 61.0%

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

       2. *-lft-identity 61.0%

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

   11. Simplified 61.0%

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

   if 3.69999999999999996e152 < 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 69.4%

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

      9. exp-neg 69.4%

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

      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 0 100.0%

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

   5. Taylor expanded in b around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. unpow2 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in b around inf 97.7%

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

      1. unpow2 97.7%

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

   10. Simplified 97.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 8: 52.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+24) (/ 1.0 (- 2.0 a)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+24) {
		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 <= 1.35d+24) 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 <= 1.35e+24) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+24:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+24)
		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 <= 1.35e+24)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+24], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e24

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-/l* 98.4%

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.3%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 74.6%

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

      9. exp-neg 74.6%

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

      10. rgt-mult-inverse 99.4%

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

      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 78.0%

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

   5. Taylor expanded in a around 0 53.0%

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

      1. neg-mul-1 53.0%

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

      2. unsub-neg 53.0%

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

   7. Simplified 53.0%

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

   if 1.35e24 < 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 68.7%

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

      9. exp-neg 68.7%

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

      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%

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

   5. Taylor expanded in b around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. unpow2 55.0%

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

   7. Simplified 55.0%

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

   8. Taylor expanded in b around inf 55.0%

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

      1. unpow2 55.0%

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

   10. Simplified 55.0%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 9: 39.3% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 73.0%

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

   9. exp-neg 73.0%

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

   10. rgt-mult-inverse 99.6%

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

   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 66.4%

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

5. Taylor expanded in a around 0 39.7%

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

   1. *-commutative 39.7%

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

7. Simplified 39.7%

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

8. Final simplification 39.7%

   \[\leadsto 0.5 + a \cdot 0.25 \]

Alternative 10: 39.9% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 73.0%

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

   9. exp-neg 73.0%

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

   10. rgt-mult-inverse 99.6%

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

   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 66.4%

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

5. Taylor expanded in a around 0 40.1%

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

   1. neg-mul-1 40.1%

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

   2. unsub-neg 40.1%

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

7. Simplified 40.1%

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

8. Final simplification 40.1%

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

Alternative 11: 39.1% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 73.0%

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

   9. exp-neg 73.0%

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

   10. rgt-mult-inverse 99.6%

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

   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 81.4%

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

5. Taylor expanded in b around 0 39.5%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 39.5%

   \[\leadsto 0.5 \]

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 2023273 
(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))))