Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 5.6s

Alternatives: 14

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% 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 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-div 97.5%

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

   4. exp-neg 97.5%

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

   5. associate-/r/ 97.5%

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

   6. /-rgt-identity 97.5%

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

   7. *-commutative 97.5%

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

   8. distribute-rgt-in 69.4%

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

   9. exp-neg 69.4%

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

   10. rgt-mult-inverse 99.1%

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

   11. prod-exp 99.8%

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

   12. unsub-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.1 \lor \neg \left(e^{b} \leq 1.00000002\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.1) (not (<= (exp b) 1.00000002)))
   (/ 1.0 (+ 1.0 (exp b)))
   (/ 1.0 (+ 1.0 (exp (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(b) <= 0.1) || !(exp(b) <= 1.00000002)) {
		tmp = 1.0 / (1.0 + exp(b));
	} else {
		tmp = 1.0 / (1.0 + exp(-a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(b) <= 0.1d0) .or. (.not. (exp(b) <= 1.00000002d0))) then
        tmp = 1.0d0 / (1.0d0 + exp(b))
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(b) <= 0.1) || !(Math.exp(b) <= 1.00000002)) {
		tmp = 1.0 / (1.0 + Math.exp(b));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(b) <= 0.1) or not (math.exp(b) <= 1.00000002):
		tmp = 1.0 / (1.0 + math.exp(b))
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((exp(b) <= 0.1) || !(exp(b) <= 1.00000002))
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(b) <= 0.1) || ~((exp(b) <= 1.00000002)))
		tmp = 1.0 / (1.0 + exp(b));
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.1], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 1.00000002]], $MachinePrecision]], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.10000000000000001 or 1.0000000200000001 < (exp.f64 b)

   1. Initial program 97.2%

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

      1. *-lft-identity 97.2%

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

      2. associate-/l* 97.2%

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

      3. remove-double-div 97.2%

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

      4. exp-neg 97.2%

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

      5. associate-/r/ 97.2%

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

      6. /-rgt-identity 97.2%

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

      7. *-commutative 97.2%

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

      8. distribute-rgt-in 78.2%

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

      9. exp-neg 78.2%

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

      10. rgt-mult-inverse 98.1%

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

      11. prod-exp 99.7%

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

      12. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around 0 99.7%

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

   if 0.10000000000000001 < (exp.f64 b) < 1.0000000200000001

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.8%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 61.4%

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

      9. exp-neg 61.5%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.1 \lor \neg \left(e^{b} \leq 1.00000002\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3: 98.6% 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 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 69.4%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   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. +-commutative 100.0%

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

      5. *-rgt-identity 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 96.6%

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

      1. *-lft-identity 96.6%

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

      2. associate-/l* 96.5%

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

      3. remove-double-div 96.5%

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

      4. exp-neg 96.5%

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

      5. associate-/r/ 96.5%

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

      6. /-rgt-identity 96.5%

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

      7. *-commutative 96.5%

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

      8. distribute-rgt-in 96.5%

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

      9. exp-neg 96.5%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 97.0%

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

4. Final simplification 97.9%

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

Alternative 4: 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 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-div 97.5%

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

   4. exp-neg 97.5%

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

   5. associate-/r/ 97.5%

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

   6. /-rgt-identity 97.5%

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

   7. *-commutative 97.5%

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

   8. distribute-rgt-in 69.4%

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

   9. exp-neg 69.4%

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

   10. rgt-mult-inverse 99.1%

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

   11. prod-exp 99.8%

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

   12. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 65.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.115) (/ (exp a) b) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = exp(a) / b;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.115d0)) then
        tmp = exp(a) / b
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.115:
		tmp = math.exp(a) / b
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.115)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.115)
		tmp = exp(a) / b;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.115], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.115000000000000005

   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 4.0%

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

      9. exp-neg 4.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 66.9%

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

      1. distribute-rgt1-in 96.3%

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

      2. rec-exp 96.3%

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

      3. associate-*r/ 96.3%

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

      4. +-commutative 96.3%

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

      5. *-rgt-identity 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in b around inf 96.3%

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

   if -0.115000000000000005 < a

   1. Initial program 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-/l* 96.5%

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

      3. remove-double-div 96.5%

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

      4. exp-neg 96.5%

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

      5. associate-/r/ 96.5%

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

      6. /-rgt-identity 96.5%

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

      7. *-commutative 96.5%

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

      8. distribute-rgt-in 96.5%

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

      9. exp-neg 96.5%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 53.4%

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

   5. Taylor expanded in a around 0 50.9%

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

      1. *-commutative 50.9%

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

   7. Simplified 50.9%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 6: 45.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.95) 0.5 (/ 1.0 (+ 2.0 (+ b (* a (- -1.0 b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 - b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.95d0)) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((-1.0d0) - b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 - b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.95:
		tmp = 0.5
	else:
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 - b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.95)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(-1.0 - b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.95)
		tmp = 0.5;
	else
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 - b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.95], 0.5, N[(1.0 / N[(2.0 + N[(b + N[(a * N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.94999999999999996

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. associate-/l* 97.6%

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

      3. remove-double-div 97.5%

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

      4. exp-neg 97.5%

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

      5. associate-/r/ 97.5%

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

      6. /-rgt-identity 97.5%

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

      7. *-commutative 97.5%

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

      8. distribute-rgt-in 97.5%

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

      9. exp-neg 97.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 18.8%

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

   if -0.94999999999999996 < b

   1. Initial program 97.5%

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

      1. *-lft-identity 97.5%

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

      2. associate-/l* 97.5%

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

      3. remove-double-div 97.5%

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

      4. exp-neg 97.5%

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

      5. associate-/r/ 97.5%

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

      6. /-rgt-identity 97.5%

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

      7. *-commutative 97.5%

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

      8. distribute-rgt-in 64.0%

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

      9. exp-neg 64.0%

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

      10. rgt-mult-inverse 98.9%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 65.4%

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

      1. distribute-rgt1-in 75.6%

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

      2. rec-exp 75.7%

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

      3. associate-*r/ 75.7%

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

      4. +-commutative 75.7%

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

      5. *-rgt-identity 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. mul-1-neg 48.6%

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

      2. unsub-neg 48.6%

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

   9. Simplified 48.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 - b\right)\right)}\\ \end{array} \]

Alternative 7: 45.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 - a\right) \cdot \left(b + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.15) 0.5 (/ 1.0 (+ 1.0 (* (- 1.0 a) (+ b 1.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.15) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (1.0 + ((1.0 - a) * (b + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.15d0)) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / (1.0d0 + ((1.0d0 - a) * (b + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.15) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (1.0 + ((1.0 - a) * (b + 1.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.15:
		tmp = 0.5
	else:
		tmp = 1.0 / (1.0 + ((1.0 - a) * (b + 1.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.15)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 - a) * Float64(b + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.15)
		tmp = 0.5;
	else
		tmp = 1.0 / (1.0 + ((1.0 - a) * (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.15], 0.5, N[(1.0 / N[(1.0 + N[(N[(1.0 - a), $MachinePrecision] * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1499999999999999

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. associate-/l* 97.6%

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

      3. remove-double-div 97.5%

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

      4. exp-neg 97.5%

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

      5. associate-/r/ 97.5%

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

      6. /-rgt-identity 97.5%

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

      7. *-commutative 97.5%

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

      8. distribute-rgt-in 97.5%

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

      9. exp-neg 97.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 18.8%

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

   if -1.1499999999999999 < b

   1. Initial program 97.5%

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

      1. *-lft-identity 97.5%

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

      2. associate-/l* 97.5%

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

      3. remove-double-div 97.5%

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

      4. exp-neg 97.5%

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

      5. associate-/r/ 97.5%

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

      6. /-rgt-identity 97.5%

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

      7. *-commutative 97.5%

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

      8. distribute-rgt-in 64.0%

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

      9. exp-neg 64.0%

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

      10. rgt-mult-inverse 98.9%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 65.4%

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

      1. distribute-rgt1-in 75.6%

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

      2. rec-exp 75.7%

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

      3. associate-*r/ 75.7%

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

      4. +-commutative 75.7%

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

      5. *-rgt-identity 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in a around 0 48.6%

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

      1. associate-+r+ 48.6%

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

      2. associate-*r* 48.6%

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

      3. distribute-rgt1-in 48.6%

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

      4. neg-mul-1 48.6%

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

   9. Simplified 48.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 - a\right) \cdot \left(b + 1\right)}\\ \end{array} \]

Alternative 8: 44.5% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.115) (/ -1.0 (* a (+ b 1.0))) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = -1.0 / (a * (b + 1.0));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.115d0)) then
        tmp = (-1.0d0) / (a * (b + 1.0d0))
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = -1.0 / (a * (b + 1.0));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.115:
		tmp = -1.0 / (a * (b + 1.0))
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.115)
		tmp = Float64(-1.0 / Float64(a * Float64(b + 1.0)));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.115)
		tmp = -1.0 / (a * (b + 1.0));
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.115], N[(-1.0 / N[(a * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.115000000000000005

   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 4.0%

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

      9. exp-neg 4.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 66.9%

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

      1. distribute-rgt1-in 96.3%

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

      2. rec-exp 96.3%

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

      3. associate-*r/ 96.3%

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

      4. +-commutative 96.3%

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

      5. *-rgt-identity 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in a around 0 24.5%

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

      1. mul-1-neg 24.5%

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

      2. unsub-neg 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in a around inf 24.5%

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

   if -0.115000000000000005 < a

   1. Initial program 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-/l* 96.5%

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

      3. remove-double-div 96.5%

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

      4. exp-neg 96.5%

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

      5. associate-/r/ 96.5%

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

      6. /-rgt-identity 96.5%

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

      7. *-commutative 96.5%

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

      8. distribute-rgt-in 96.5%

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

      9. exp-neg 96.5%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 53.4%

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

   5. Taylor expanded in a around 0 50.9%

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

      1. *-commutative 50.9%

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

   7. Simplified 50.9%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9: 44.7% accurate, 33.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-/l* 97.7%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 70.2%

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

      9. exp-neg 70.2%

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

      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 b around 0 81.0%

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

   5. Taylor expanded in a around 0 52.0%

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

      1. neg-mul-1 52.0%

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

      2. unsub-neg 52.0%

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

   7. Simplified 52.0%

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

   if 2 < b

   1. Initial program 97.0%

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

      1. *-lft-identity 97.0%

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

      2. associate-/l* 97.0%

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

      3. remove-double-div 97.0%

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

      4. exp-neg 97.0%

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

      5. associate-/r/ 97.0%

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

      6. /-rgt-identity 97.0%

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

      7. *-commutative 97.0%

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

      8. distribute-rgt-in 67.5%

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

      9. exp-neg 67.5%

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

      10. rgt-mult-inverse 97.0%

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

      11. prod-exp 99.6%

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

      12. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in b around 0 33.3%

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

      1. distribute-rgt1-in 33.3%

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

      2. rec-exp 33.3%

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

      3. associate-*r/ 33.3%

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

      4. +-commutative 33.3%

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

      5. *-rgt-identity 33.3%

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

   6. Simplified 33.3%

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

   7. Taylor expanded in a around 0 23.7%

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

      1. mul-1-neg 23.7%

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

      2. unsub-neg 23.7%

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

   9. Simplified 23.7%

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

   10. Taylor expanded in b around inf 23.7%

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

4. Final simplification 43.4%

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

Alternative 10: 43.1% accurate, 33.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-div 97.5%

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

   4. exp-neg 97.5%

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

   5. associate-/r/ 97.5%

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

   6. /-rgt-identity 97.5%

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

   7. *-commutative 97.5%

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

   8. distribute-rgt-in 69.4%

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

   9. exp-neg 69.4%

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

   10. rgt-mult-inverse 99.1%

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

   11. prod-exp 99.8%

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

   12. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in b around 0 55.7%

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

   1. distribute-rgt1-in 64.3%

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

   2. rec-exp 64.3%

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

   3. associate-*r/ 64.3%

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

   4. +-commutative 64.3%

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

   5. *-rgt-identity 64.3%

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

6. Simplified 64.3%

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

7. Taylor expanded in a around 0 41.2%

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

   1. mul-1-neg 41.2%

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

   2. unsub-neg 41.2%

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

9. Simplified 41.2%

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

10. Taylor expanded in a around inf 42.0%

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

    1. mul-1-neg 42.0%

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

    2. distribute-rgt-neg-out 42.0%

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

    3. distribute-neg-in 42.0%

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

    4. metadata-eval 42.0%

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

    5. unsub-neg 42.0%

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

12. Simplified 42.0%

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

13. Final simplification 42.0%

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

Alternative 11: 44.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.115) (/ -1.0 (* b a)) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = -1.0 / (b * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.115d0)) then
        tmp = (-1.0d0) / (b * a)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.115) {
		tmp = -1.0 / (b * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.115:
		tmp = -1.0 / (b * a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.115)
		tmp = Float64(-1.0 / Float64(b * a));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.115)
		tmp = -1.0 / (b * a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.115], N[(-1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.115000000000000005

   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 4.0%

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

      9. exp-neg 4.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 66.9%

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

      1. distribute-rgt1-in 96.3%

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

      2. rec-exp 96.3%

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

      3. associate-*r/ 96.3%

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

      4. +-commutative 96.3%

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

      5. *-rgt-identity 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in a around 0 24.5%

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

      1. mul-1-neg 24.5%

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

      2. unsub-neg 24.5%

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

   9. Simplified 24.5%

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

   10. Taylor expanded in a around inf 24.5%

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

       1. mul-1-neg 24.5%

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

       2. distribute-rgt-neg-out 24.5%

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

       3. distribute-neg-in 24.5%

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

       4. metadata-eval 24.5%

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

       5. unsub-neg 24.5%

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

   12. Simplified 24.5%

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

   13. Taylor expanded in b around inf 23.2%

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

   if -0.115000000000000005 < a

   1. Initial program 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-/l* 96.5%

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

      3. remove-double-div 96.5%

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

      4. exp-neg 96.5%

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

      5. associate-/r/ 96.5%

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

      6. /-rgt-identity 96.5%

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

      7. *-commutative 96.5%

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

      8. distribute-rgt-in 96.5%

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

      9. exp-neg 96.5%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 53.4%

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

   5. Taylor expanded in a around 0 50.9%

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

      1. *-commutative 50.9%

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

   7. Simplified 50.9%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.115:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 12: 44.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.3e+90) (/ 1.0 (- 2.0 a)) (/ -1.0 (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.3e+90) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (b * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.3d+90) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-1.0d0) / (b * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.3e+90) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.3e+90:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -1.0 / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.3e+90)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-1.0 / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.3e+90)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -1.0 / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.3e+90], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.30000000000000008e90

   1. Initial program 97.4%

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

      1. *-lft-identity 97.4%

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

      2. associate-/l* 97.4%

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

      3. remove-double-div 97.3%

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

      4. exp-neg 97.3%

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

      5. associate-/r/ 97.3%

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

      6. /-rgt-identity 97.3%

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

      7. *-commutative 97.3%

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

      8. distribute-rgt-in 70.2%

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

      9. exp-neg 70.3%

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

      10. rgt-mult-inverse 99.3%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 74.3%

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

   5. Taylor expanded in a around 0 46.0%

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

      1. neg-mul-1 46.0%

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

      2. unsub-neg 46.0%

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

   7. Simplified 46.0%

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

   if 3.30000000000000008e90 < b

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-/l* 98.1%

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

      3. remove-double-div 98.1%

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

      4. exp-neg 98.1%

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

      5. associate-/r/ 98.1%

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

      6. /-rgt-identity 98.1%

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

      7. *-commutative 98.1%

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

      8. distribute-rgt-in 66.0%

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

      9. exp-neg 66.0%

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

      10. rgt-mult-inverse 98.1%

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

      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 36.5%

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

      1. distribute-rgt1-in 36.5%

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

      2. rec-exp 36.5%

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

      3. associate-*r/ 36.5%

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

      4. +-commutative 36.5%

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

      5. *-rgt-identity 36.5%

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

   6. Simplified 36.5%

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

   7. Taylor expanded in a around 0 33.1%

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

      1. mul-1-neg 33.1%

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

      2. unsub-neg 33.1%

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

   9. Simplified 33.1%

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

   10. Taylor expanded in a around inf 31.6%

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

       1. mul-1-neg 31.6%

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

       2. distribute-rgt-neg-out 31.6%

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

       3. distribute-neg-in 31.6%

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

       4. metadata-eval 31.6%

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

       5. unsub-neg 31.6%

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

   12. Simplified 31.6%

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

   13. Taylor expanded in b around inf 31.6%

       \[\leadsto \color{blue}{\frac{-1}{a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \end{array} \]

Alternative 13: 39.4% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-div 97.5%

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

   4. exp-neg 97.5%

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

   5. associate-/r/ 97.5%

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

   6. /-rgt-identity 97.5%

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

   7. *-commutative 97.5%

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

   8. distribute-rgt-in 69.4%

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

   9. exp-neg 69.4%

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

   10. rgt-mult-inverse 99.1%

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

   11. prod-exp 99.8%

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

   12. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in b around 0 66.0%

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

5. Taylor expanded in a around 0 36.7%

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

   1. *-commutative 36.7%

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

7. Simplified 36.7%

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

8. Final simplification 36.7%

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

Alternative 14: 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 97.5%

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

   1. *-lft-identity 97.5%

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

   2. associate-/l* 97.5%

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

   3. remove-double-div 97.5%

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

   4. exp-neg 97.5%

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

   5. associate-/r/ 97.5%

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

   6. /-rgt-identity 97.5%

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

   7. *-commutative 97.5%

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

   8. distribute-rgt-in 69.4%

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

   9. exp-neg 69.4%

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

   10. rgt-mult-inverse 99.1%

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

   11. prod-exp 99.8%

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

   12. unsub-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in a around 0 79.3%

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

5. Taylor expanded in b around 0 36.5%

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

6. Final simplification 36.5%

   \[\leadsto 0.5 \]

Developer target: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(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))))