Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 7.8s

Alternatives: 12

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{b - a}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- (log1p (exp (- b a))))))

C

double code(double a, double b) {
	return exp(-log1p(exp((b - a))));
}

Java

public static double code(double a, double b) {
	return Math.exp(-Math.log1p(Math.exp((b - a))));
}

Python

def code(a, b):
	return math.exp(-math.log1p(math.exp((b - a))))

Julia

function code(a, b)
	return exp(Float64(-log1p(exp(Float64(b - a)))))
end

Wolfram

code[a_, b_] := N[Exp[(-N[Log[1 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-/l* 99.6%

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

   3. remove-double-div 99.6%

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

   4. exp-neg 99.6%

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

   5. associate-/r/ 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. distribute-rgt-in 73.4%

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

   9. exp-neg 73.4%

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

   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. Step-by-step derivation

   1. add-exp-log 100.0%

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

   2. log-rec 100.0%

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

   3. log1p-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.5:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.5

   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 2.8%

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

      9. exp-neg 2.8%

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

      10. rgt-mult-inverse 98.5%

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

      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 0 100.0%

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

   if 0.5 < (exp.f64 a)

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.5:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		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 (exp(a) <= 0.0)
		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 (exp(a) <= 0.0)
		tmp = exp(a) / b;
	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], 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 (exp.f64 a) < 0.0

   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 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 98.5%

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

      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 72.1%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.4%

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

4. Final simplification 98.9%

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

Alternative 4: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ (exp a) b)
   (/ 1.0 (+ 2.0 (/ (* b (- 1.0 (* (* b b) 0.25))) (+ 1.0 (* b -0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 - ((b * b) * 0.25d0))) / (1.0d0 + (b * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 - N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   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 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 98.5%

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

      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 72.1%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.4%

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

   5. Taylor expanded in b around 0 67.4%

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

      1. unpow2 67.4%

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

   7. Simplified 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. associate-*r* 67.4%

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

      3. distribute-rgt-out 67.4%

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

      4. *-commutative 67.4%

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

   9. Applied egg-rr 67.4%

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

       1. *-commutative 67.4%

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

       2. flip-+ 67.4%

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

       3. associate-*l/ 70.8%

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

       4. metadata-eval 70.8%

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

       5. swap-sqr 70.8%

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

       6. metadata-eval 70.8%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

       7. *-commutative 70.8%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

       8. cancel-sign-sub-inv 70.8%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

       9. metadata-eval 70.8%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   11. Applied egg-rr 70.8%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \]

Alternative 5: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-/l* 99.6%

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

   3. remove-double-div 99.6%

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

   4. exp-neg 99.6%

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

   5. associate-/r/ 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. distribute-rgt-in 73.4%

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

   9. exp-neg 73.4%

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

   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%

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

Alternative 6: 69.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot 0.5\right)\\ \mathbf{if}\;b \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot b - t_0 \cdot t_0}{b - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b 0.5))))
   (if (<= b 1.65e+77)
     (/ 1.0 (+ 2.0 (- (* a (* a 0.5)) a)))
     (if (<= b 1e+103)
       (/ 1.0 (+ 2.0 (/ (- (* b b) (* t_0 t_0)) (- b t_0))))
       (/
        1.0
        (+ 2.0 (/ (* b (- 1.0 (* (* b b) 0.25))) (+ 1.0 (* b -0.5)))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * 0.5);
	double tmp;
	if (b <= 1.65e+77) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * 0.5d0)
    if (b <= 1.65d+77) then
        tmp = 1.0d0 / (2.0d0 + ((a * (a * 0.5d0)) - a))
    else if (b <= 1d+103) then
        tmp = 1.0d0 / (2.0d0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
    else
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 - ((b * b) * 0.25d0))) / (1.0d0 + (b * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * 0.5);
	double tmp;
	if (b <= 1.65e+77) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * 0.5)
	tmp = 0
	if b <= 1.65e+77:
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a))
	elif b <= 1e+103:
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
	else:
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * 0.5))
	tmp = 0.0
	if (b <= 1.65e+77)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(a * Float64(a * 0.5)) - a)));
	elseif (b <= 1e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(b - t_0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(b * Float64(1.0 - Float64(Float64(b * b) * 0.25))) / Float64(1.0 + Float64(b * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * 0.5);
	tmp = 0.0;
	if (b <= 1.65e+77)
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	elseif (b <= 1e+103)
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	else
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.65e+77], N[(1.0 / N[(2.0 + N[(N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+103], N[(1.0 / N[(2.0 + N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 - N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6499999999999999e77

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. remove-double-div 99.5%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 76.6%

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

      9. exp-neg 76.6%

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

      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 77.3%

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

   5. Taylor expanded in a around 0 62.2%

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

      1. +-commutative 62.2%

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

      2. neg-mul-1 62.2%

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

      3. unsub-neg 62.2%

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

      4. *-commutative 62.2%

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

      5. unpow2 62.2%

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

      6. associate-*l* 62.2%

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

   7. Simplified 62.2%

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

   if 1.6499999999999999e77 < b < 1e103

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 60.0%

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

      9. exp-neg 60.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 5.0%

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

      1. unpow2 5.0%

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

   7. Simplified 5.0%

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

      1. *-un-lft-identity 5.0%

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

      2. associate-*r* 5.0%

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

      3. distribute-rgt-out 5.0%

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

      4. *-commutative 5.0%

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

   9. Applied egg-rr 5.0%

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

       1. distribute-lft-in 5.0%

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

       2. *-rgt-identity 5.0%

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

       3. flip-+ 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 1e103 < 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 63.0%

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

      9. exp-neg 63.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 76.1%

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

      1. unpow2 76.1%

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

   7. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. associate-*r* 76.1%

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

      3. distribute-rgt-out 76.1%

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

      4. *-commutative 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. *-commutative 76.1%

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

       2. flip-+ 76.1%

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

       3. associate-*l/ 100.0%

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

       4. metadata-eval 100.0%

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

       5. swap-sqr 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

       7. *-commutative 100.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

       8. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

       9. metadata-eval 100.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot b - \left(b \cdot \left(b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(b \cdot 0.5\right)\right)}{b - b \cdot \left(b \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \]

Alternative 7: 67.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e+87)
   (/ 1.0 (+ 2.0 (- (* a (* a 0.5)) a)))
   (/ 1.0 (+ 2.0 (/ (* b (- 1.0 (* (* b b) 0.25))) (+ 1.0 (* b -0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e+87) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 4d+87) then
        tmp = 1.0d0 / (2.0d0 + ((a * (a * 0.5d0)) - a))
    else
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 - ((b * b) * 0.25d0))) / (1.0d0 + (b * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4e+87) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4e+87:
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a))
	else:
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e+87)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(a * Float64(a * 0.5)) - a)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(b * Float64(1.0 - Float64(Float64(b * b) * 0.25))) / Float64(1.0 + Float64(b * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4e+87)
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	else
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e+87], N[(1.0 / N[(2.0 + N[(N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 - N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9999999999999998e87

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. remove-double-div 99.5%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 76.3%

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

      9. exp-neg 76.4%

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

      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 b around 0 77.0%

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

   5. Taylor expanded in a around 0 61.7%

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

      1. +-commutative 61.7%

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

      2. neg-mul-1 61.7%

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

      3. unsub-neg 61.7%

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

      4. *-commutative 61.7%

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

      5. unpow2 61.7%

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

      6. associate-*l* 61.7%

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

   7. Simplified 61.7%

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

   if 3.9999999999999998e87 < 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 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%

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

   5. Taylor expanded in b around 0 72.4%

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

      1. unpow2 72.4%

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

   7. Simplified 72.4%

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

      1. *-un-lft-identity 72.4%

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

      2. associate-*r* 72.4%

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

      3. distribute-rgt-out 72.4%

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

      4. *-commutative 72.4%

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

   9. Applied egg-rr 72.4%

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

       1. *-commutative 72.4%

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

       2. flip-+ 72.4%

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

       3. associate-*l/ 95.0%

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

       4. metadata-eval 95.0%

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

       5. swap-sqr 95.0%

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

       6. metadata-eval 95.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

       7. *-commutative 95.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

       8. cancel-sign-sub-inv 95.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

       9. metadata-eval 95.0%

          \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   11. Applied egg-rr 95.0%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \]

Alternative 8: 64.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e+149) (/ 1.0 (+ 2.0 (- (* a (* a 0.5)) a))) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+149) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - 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 <= 6.6d+149) then
        tmp = 1.0d0 / (2.0d0 + ((a * (a * 0.5d0)) - 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 <= 6.6e+149) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e+149:
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e+149)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(a * Float64(a * 0.5)) - 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 <= 6.6e+149)
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e+149], N[(1.0 / N[(2.0 + N[(N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6e149

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. remove-double-div 99.5%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 74.4%

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

      9. exp-neg 74.4%

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

      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 b around 0 75.1%

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

   5. Taylor expanded in a around 0 59.6%

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

      1. +-commutative 59.6%

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

      2. neg-mul-1 59.6%

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

      3. unsub-neg 59.6%

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

      4. *-commutative 59.6%

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

      5. unpow2 59.6%

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

      6. associate-*l* 59.6%

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

   7. Simplified 59.6%

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

   if 6.6e149 < 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 68.3%

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

      9. exp-neg 68.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 97.8%

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

      1. unpow2 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in b around inf 97.8%

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

      1. unpow2 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + \left(a \cdot \left(a \cdot 0.5\right) - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 9: 63.9% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+146}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e+146) (/ (+ a 2.0) (- 4.0 (* a a))) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.75e+146) {
		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 <= 1.75d+146) 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 <= 1.75e+146) {
		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 <= 1.75e+146:
		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 <= 1.75e+146)
		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 <= 1.75e+146)
		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, 1.75e+146], 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 < 1.7500000000000001e146

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. remove-double-div 99.5%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 74.4%

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

      9. exp-neg 74.4%

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

      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 b around 0 75.1%

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

   5. Taylor expanded in a around 0 49.7%

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

      1. neg-mul-1 49.7%

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

      2. unsub-neg 49.7%

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

   7. Simplified 49.7%

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

      1. flip-- 59.2%

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

      2. associate-/r/ 59.2%

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

      3. metadata-eval 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. associate-*l/ 59.2%

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

       2. *-lft-identity 59.2%

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

   11. Simplified 59.2%

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

   if 1.7500000000000001e146 < 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 68.3%

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

      9. exp-neg 68.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 97.8%

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

      1. unpow2 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in b around inf 97.8%

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

      1. unpow2 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{+146}:\\ \;\;\;\;\frac{a + 2}{4 - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 10: 54.0% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.45e+75) (/ 1.0 (- 2.0 a)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.45e+75) {
		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 <= 1.45d+75) 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 <= 1.45e+75) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.45e+75:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.45e+75)
		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 <= 1.45e+75)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.45e+75], 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 < 1.4499999999999999e75

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-/l* 99.5%

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

      3. remove-double-div 99.5%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 76.6%

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

      9. exp-neg 76.6%

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

      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 77.3%

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

   5. Taylor expanded in a around 0 53.9%

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

      1. neg-mul-1 53.9%

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

      2. unsub-neg 53.9%

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

   7. Simplified 53.9%

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

   if 1.4499999999999999e75 < 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 62.7%

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

      9. exp-neg 62.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 70.1%

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

      1. unpow2 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in b around inf 70.1%

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

      1. unpow2 70.1%

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

   10. Simplified 70.1%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 11: 40.0% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-/l* 99.6%

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

   3. remove-double-div 99.6%

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

   4. exp-neg 99.6%

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

   5. associate-/r/ 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. distribute-rgt-in 73.4%

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

   9. exp-neg 73.4%

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

   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 68.5%

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

5. Taylor expanded in a around 0 42.4%

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

   1. neg-mul-1 42.4%

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

   2. unsub-neg 42.4%

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

7. Simplified 42.4%

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

8. Final simplification 42.4%

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

Alternative 12: 39.2% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-/l* 99.6%

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

   3. remove-double-div 99.6%

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

   4. exp-neg 99.6%

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

   5. associate-/r/ 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. distribute-rgt-in 73.4%

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

   9. exp-neg 73.4%

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

   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 84.1%

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

5. Taylor expanded in b around 0 41.7%

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

6. Final simplification 41.7%

   \[\leadsto 0.5 \]

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 2023282 
(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))))