Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 6.1s

Alternatives: 10

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% 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. add-exp-log 98.8%

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

   2. div-exp 98.8%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in a around -inf 98.8%

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

   1. exp-neg 98.8%

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

   2. neg-mul-1 98.8%

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

   3. prod-exp 98.8%

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

   4. rem-exp-log 98.8%

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

   5. *-commutative 98.8%

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

   6. distribute-rgt-in 74.2%

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

   7. rec-exp 74.2%

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

   8. rgt-mult-inverse 100.0%

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

   9. prod-exp 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) 0.0 (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = 0.0;
	} 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 (exp(a) <= 0.0d0) then
        tmp = 0.0d0
    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 (Math.exp(a) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], 0.0, N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 0.0%

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

      7. rec-exp 0.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 81.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-188}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-208}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-146}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.9)
   1.0
   (if (<= b -5.4e-188)
     (+ 0.5 (* b -0.25))
     (if (<= b -1.25e-208)
       (exp a)
       (if (<= b 1.08e-157)
         (+ 0.5 (* a 0.25))
         (if (<= b 5e-146) (exp a) (if (<= b 38.0) (/ 1.0 (+ b 2.0)) 0.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.9) {
		tmp = 1.0;
	} else if (b <= -5.4e-188) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= -1.25e-208) {
		tmp = exp(a);
	} else if (b <= 1.08e-157) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 5e-146) {
		tmp = exp(a);
	} else if (b <= 38.0) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.9d0)) then
        tmp = 1.0d0
    else if (b <= (-5.4d-188)) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else if (b <= (-1.25d-208)) then
        tmp = exp(a)
    else if (b <= 1.08d-157) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 5d-146) then
        tmp = exp(a)
    else if (b <= 38.0d0) then
        tmp = 1.0d0 / (b + 2.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.9) {
		tmp = 1.0;
	} else if (b <= -5.4e-188) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= -1.25e-208) {
		tmp = Math.exp(a);
	} else if (b <= 1.08e-157) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 5e-146) {
		tmp = Math.exp(a);
	} else if (b <= 38.0) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.9:
		tmp = 1.0
	elif b <= -5.4e-188:
		tmp = 0.5 + (b * -0.25)
	elif b <= -1.25e-208:
		tmp = math.exp(a)
	elif b <= 1.08e-157:
		tmp = 0.5 + (a * 0.25)
	elif b <= 5e-146:
		tmp = math.exp(a)
	elif b <= 38.0:
		tmp = 1.0 / (b + 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.9)
		tmp = 1.0;
	elseif (b <= -5.4e-188)
		tmp = Float64(0.5 + Float64(b * -0.25));
	elseif (b <= -1.25e-208)
		tmp = exp(a);
	elseif (b <= 1.08e-157)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 5e-146)
		tmp = exp(a);
	elseif (b <= 38.0)
		tmp = Float64(1.0 / Float64(b + 2.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.9)
		tmp = 1.0;
	elseif (b <= -5.4e-188)
		tmp = 0.5 + (b * -0.25);
	elseif (b <= -1.25e-208)
		tmp = exp(a);
	elseif (b <= 1.08e-157)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 5e-146)
		tmp = exp(a);
	elseif (b <= 38.0)
		tmp = 1.0 / (b + 2.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.9], 1.0, If[LessEqual[b, -5.4e-188], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.25e-208], N[Exp[a], $MachinePrecision], If[LessEqual[b, 1.08e-157], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-146], N[Exp[a], $MachinePrecision], If[LessEqual[b, 38.0], N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 6 regimes

2. if b < -1.8999999999999999

   1. Initial program 98.1%

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

      1. add-exp-log 98.1%

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

      2. div-exp 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in a around -inf 98.1%

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

      1. exp-neg 98.1%

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

      2. neg-mul-1 98.1%

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

      3. prod-exp 98.0%

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

      4. rem-exp-log 98.0%

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

      5. *-commutative 98.0%

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

      6. distribute-rgt-in 98.0%

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

      7. rec-exp 98.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 20.3%

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

   9. Applied egg-rr 100.0%

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

   if -1.8999999999999999 < b < -5.4000000000000002e-188

   1. Initial program 93.5%

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

   2. Taylor expanded in a around 0 72.9%

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

   3. Taylor expanded in b around 0 72.9%

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

      1. *-commutative 72.9%

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

   5. Simplified 72.9%

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

   if -5.4000000000000002e-188 < b < -1.24999999999999991e-208 or 1.0799999999999999e-157 < b < 4.99999999999999957e-146

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 87.5%

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

   if -1.24999999999999991e-208 < b < 1.0799999999999999e-157

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

   if 4.99999999999999957e-146 < b < 38

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 86.2%

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

   3. Taylor expanded in b around 0 83.7%

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

      1. +-commutative 83.7%

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

   5. Simplified 83.7%

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

   if 38 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 64.8%

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

      7. rec-exp 64.8%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 37.2%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-188}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-208}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-146}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 80.6% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-158} \lor \neg \left(b \leq 5 \cdot 10^{-146}\right) \land b \leq 38:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8.2e-13)
   1.0
   (if (or (<= b 9.8e-158) (and (not (<= b 5e-146)) (<= b 38.0)))
     (+ 0.5 (* a 0.25))
     0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if ((b <= 9.8e-158) || (!(b <= 5e-146) && (b <= 38.0))) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8.2d-13)) then
        tmp = 1.0d0
    else if ((b <= 9.8d-158) .or. (.not. (b <= 5d-146)) .and. (b <= 38.0d0)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if ((b <= 9.8e-158) || (!(b <= 5e-146) && (b <= 38.0))) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8.2e-13:
		tmp = 1.0
	elif (b <= 9.8e-158) or (not (b <= 5e-146) and (b <= 38.0)):
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif ((b <= 9.8e-158) || (!(b <= 5e-146) && (b <= 38.0)))
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif ((b <= 9.8e-158) || (~((b <= 5e-146)) && (b <= 38.0)))
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8.2e-13], 1.0, If[Or[LessEqual[b, 9.8e-158], And[N[Not[LessEqual[b, 5e-146]], $MachinePrecision], LessEqual[b, 38.0]]], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000004e-13

   1. Initial program 96.4%

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

      1. add-exp-log 96.4%

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

      2. div-exp 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in a around -inf 96.4%

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

      1. exp-neg 96.5%

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

      2. neg-mul-1 96.5%

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

      3. prod-exp 96.3%

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

      4. rem-exp-log 96.3%

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

      5. *-commutative 96.3%

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

      6. distribute-rgt-in 94.5%

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

      7. rec-exp 94.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 24.2%

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

   9. Applied egg-rr 96.8%

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

   if -8.2000000000000004e-13 < b < 9.79999999999999986e-158 or 4.99999999999999957e-146 < b < 38

   1. Initial program 99.1%

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

   2. Taylor expanded in b around 0 97.7%

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

   3. Taylor expanded in a around 0 72.4%

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

      1. *-commutative 72.4%

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

   5. Simplified 72.4%

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

   if 9.79999999999999986e-158 < b < 4.99999999999999957e-146 or 38 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 60.3%

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

      7. rec-exp 60.3%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 42.9%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-158} \lor \neg \left(b \leq 5 \cdot 10^{-146}\right) \land b \leq 38:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 80.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.047:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8.2e-13)
   1.0
   (if (<= b 1.08e-157)
     (+ 0.5 (* a 0.25))
     (if (<= b 6.8e-149) 0.0 (if (<= b 0.047) (+ 0.5 (* b -0.25)) 0.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 1.08e-157) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 0.047) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8.2d-13)) then
        tmp = 1.0d0
    else if (b <= 1.08d-157) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 6.8d-149) then
        tmp = 0.0d0
    else if (b <= 0.047d0) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 1.08e-157) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 0.047) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8.2e-13:
		tmp = 1.0
	elif b <= 1.08e-157:
		tmp = 0.5 + (a * 0.25)
	elif b <= 6.8e-149:
		tmp = 0.0
	elif b <= 0.047:
		tmp = 0.5 + (b * -0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 1.08e-157)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 0.047)
		tmp = Float64(0.5 + Float64(b * -0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 1.08e-157)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 0.047)
		tmp = 0.5 + (b * -0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8.2e-13], 1.0, If[LessEqual[b, 1.08e-157], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-149], 0.0, If[LessEqual[b, 0.047], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.2000000000000004e-13

   1. Initial program 96.4%

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

      1. add-exp-log 96.4%

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

      2. div-exp 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in a around -inf 96.4%

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

      1. exp-neg 96.5%

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

      2. neg-mul-1 96.5%

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

      3. prod-exp 96.3%

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

      4. rem-exp-log 96.3%

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

      5. *-commutative 96.3%

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

      6. distribute-rgt-in 94.5%

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

      7. rec-exp 94.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 24.2%

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

   9. Applied egg-rr 96.8%

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

   if -8.2000000000000004e-13 < b < 1.0799999999999999e-157

   1. Initial program 98.8%

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

   2. Taylor expanded in b around 0 98.8%

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

   3. Taylor expanded in a around 0 67.9%

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

      1. *-commutative 67.9%

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

   5. Simplified 67.9%

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

   if 1.0799999999999999e-157 < b < 6.7999999999999998e-149 or 0.047 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 61.0%

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

      7. rec-exp 61.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 41.0%

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

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{0} \]

   if 6.7999999999999998e-149 < b < 0.047

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 84.0%

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

   3. Taylor expanded in b around 0 84.1%

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

      1. *-commutative 84.1%

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

   5. Simplified 84.1%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.047:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 80.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-158}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8.2e-13)
   1.0
   (if (<= b 8.8e-158)
     (+ 0.5 (* a 0.25))
     (if (<= b 7.5e-149) 0.0 (if (<= b 38.0) (/ 1.0 (+ b 2.0)) 0.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 8.8e-158) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 7.5e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8.2d-13)) then
        tmp = 1.0d0
    else if (b <= 8.8d-158) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 7.5d-149) then
        tmp = 0.0d0
    else if (b <= 38.0d0) then
        tmp = 1.0d0 / (b + 2.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 8.8e-158) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 7.5e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 1.0 / (b + 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8.2e-13:
		tmp = 1.0
	elif b <= 8.8e-158:
		tmp = 0.5 + (a * 0.25)
	elif b <= 7.5e-149:
		tmp = 0.0
	elif b <= 38.0:
		tmp = 1.0 / (b + 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 8.8e-158)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 7.5e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = Float64(1.0 / Float64(b + 2.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 8.8e-158)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 7.5e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = 1.0 / (b + 2.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8.2e-13], 1.0, If[LessEqual[b, 8.8e-158], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-149], 0.0, If[LessEqual[b, 38.0], N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.2000000000000004e-13

   1. Initial program 96.4%

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

      1. add-exp-log 96.4%

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

      2. div-exp 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in a around -inf 96.4%

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

      1. exp-neg 96.5%

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

      2. neg-mul-1 96.5%

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

      3. prod-exp 96.3%

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

      4. rem-exp-log 96.3%

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

      5. *-commutative 96.3%

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

      6. distribute-rgt-in 94.5%

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

      7. rec-exp 94.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 24.2%

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

   9. Applied egg-rr 96.8%

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

   if -8.2000000000000004e-13 < b < 8.8000000000000004e-158

   1. Initial program 98.8%

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

   2. Taylor expanded in b around 0 98.8%

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

   3. Taylor expanded in a around 0 67.9%

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

      1. *-commutative 67.9%

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

   5. Simplified 67.9%

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

   if 8.8000000000000004e-158 < b < 7.49999999999999995e-149 or 38 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 60.5%

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

      7. rec-exp 60.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 41.4%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]

   if 7.49999999999999995e-149 < b < 38

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 84.4%

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

   3. Taylor expanded in b around 0 82.1%

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

      1. +-commutative 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-158}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\frac{1}{b + 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 80.4% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8.2e-13)
   1.0
   (if (<= b 1.08e-157)
     0.5
     (if (<= b 6.8e-149) 0.0 (if (<= b 38.0) 0.5 0.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 1.08e-157) {
		tmp = 0.5;
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8.2d-13)) then
        tmp = 1.0d0
    else if (b <= 1.08d-157) then
        tmp = 0.5d0
    else if (b <= 6.8d-149) then
        tmp = 0.0d0
    else if (b <= 38.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8.2e-13) {
		tmp = 1.0;
	} else if (b <= 1.08e-157) {
		tmp = 0.5;
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8.2e-13:
		tmp = 1.0
	elif b <= 1.08e-157:
		tmp = 0.5
	elif b <= 6.8e-149:
		tmp = 0.0
	elif b <= 38.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 1.08e-157)
		tmp = 0.5;
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8.2e-13)
		tmp = 1.0;
	elseif (b <= 1.08e-157)
		tmp = 0.5;
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8.2e-13], 1.0, If[LessEqual[b, 1.08e-157], 0.5, If[LessEqual[b, 6.8e-149], 0.0, If[LessEqual[b, 38.0], 0.5, 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000004e-13

   1. Initial program 96.4%

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

      1. add-exp-log 96.4%

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

      2. div-exp 96.4%

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

   3. Applied egg-rr 96.4%

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

   4. Taylor expanded in a around -inf 96.4%

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

      1. exp-neg 96.5%

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

      2. neg-mul-1 96.5%

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

      3. prod-exp 96.3%

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

      4. rem-exp-log 96.3%

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

      5. *-commutative 96.3%

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

      6. distribute-rgt-in 94.5%

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

      7. rec-exp 94.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 24.2%

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

   9. Applied egg-rr 96.8%

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

   if -8.2000000000000004e-13 < b < 1.0799999999999999e-157 or 6.7999999999999998e-149 < b < 38

   1. Initial program 99.1%

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

   2. Taylor expanded in a around 0 73.0%

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

   3. Taylor expanded in b around 0 71.6%

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

   if 1.0799999999999999e-157 < b < 6.7999999999999998e-149 or 38 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 60.5%

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

      7. rec-exp 60.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 41.4%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 65.9% accurate, 42.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-157) 0.5 (if (<= b 6.8e-149) 0.0 (if (<= b 38.0) 0.5 0.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-157) {
		tmp = 0.5;
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.08d-157) then
        tmp = 0.5d0
    else if (b <= 6.8d-149) then
        tmp = 0.0d0
    else if (b <= 38.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-157) {
		tmp = 0.5;
	} else if (b <= 6.8e-149) {
		tmp = 0.0;
	} else if (b <= 38.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.08e-157:
		tmp = 0.5
	elif b <= 6.8e-149:
		tmp = 0.0
	elif b <= 38.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-157)
		tmp = 0.5;
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-157)
		tmp = 0.5;
	elseif (b <= 6.8e-149)
		tmp = 0.0;
	elseif (b <= 38.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.08e-157], 0.5, If[LessEqual[b, 6.8e-149], 0.0, If[LessEqual[b, 38.0], 0.5, 0.0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0799999999999999e-157 or 6.7999999999999998e-149 < b < 38

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0 80.3%

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

   3. Taylor expanded in b around 0 55.7%

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

   if 1.0799999999999999e-157 < b < 6.7999999999999998e-149 or 38 < b

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around -inf 100.0%

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

      1. exp-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. prod-exp 100.0%

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

      4. rem-exp-log 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-in 60.5%

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

      7. rec-exp 60.5%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 41.4%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-157}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 48.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -900000:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -900000.0) 0.1111111111111111 0.0))

C

double code(double a, double b) {
	double tmp;
	if (b <= -900000.0) {
		tmp = 0.1111111111111111;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-900000.0d0)) then
        tmp = 0.1111111111111111d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -900000.0) {
		tmp = 0.1111111111111111;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -900000.0:
		tmp = 0.1111111111111111
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -900000.0)
		tmp = 0.1111111111111111;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -900000.0)
		tmp = 0.1111111111111111;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -900000.0], 0.1111111111111111, 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -9e5

   1. Initial program 98.1%

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

      1. add-exp-log 98.1%

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

      2. div-exp 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in a around -inf 98.1%

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

      1. exp-neg 98.1%

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

      2. neg-mul-1 98.1%

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

      3. prod-exp 98.0%

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

      4. rem-exp-log 98.0%

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

      5. *-commutative 98.0%

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

      6. distribute-rgt-in 98.0%

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

      7. rec-exp 98.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 20.3%

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

   9. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{0.1111111111111111} \]

   if -9e5 < b

   1. Initial program 99.0%

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

      1. add-exp-log 99.0%

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

      2. div-exp 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in a around -inf 99.0%

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

      1. exp-neg 99.0%

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

      2. neg-mul-1 99.0%

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

      3. prod-exp 99.0%

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

      4. rem-exp-log 99.0%

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

      5. *-commutative 99.0%

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

      6. distribute-rgt-in 68.1%

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

      7. rec-exp 68.1%

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

      8. rgt-mult-inverse 100.0%

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

      9. prod-exp 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 77.1%

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

   9. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -900000:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 46.6% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.0)

C

double code(double a, double b) {
	return 0.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

public static double code(double a, double b) {
	return 0.0;
}

Python

def code(a, b):
	return 0.0

Julia

function code(a, b)
	return 0.0
end

MATLAB

function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

code[a_, b_] := 0.0

Derivation

1. Initial program 98.8%

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

   1. add-exp-log 98.8%

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

   2. div-exp 98.8%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in a around -inf 98.8%

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

   1. exp-neg 98.8%

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

   2. neg-mul-1 98.8%

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

   3. prod-exp 98.8%

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

   4. rem-exp-log 98.8%

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

   5. *-commutative 98.8%

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

   6. distribute-rgt-in 74.2%

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

   7. rec-exp 74.2%

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

   8. rgt-mult-inverse 100.0%

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

   9. prod-exp 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in b around 0 65.6%

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

9. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 44.4%

    \[\leadsto 0 \]

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 2023321 
(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))))