Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 10.8s

Alternatives: 21

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + (exp(1) ^ Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + (2.71828182845904523536 ^ (b - a)));
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 75.4%

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

   7. *-lft-identity 75.4%

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

   8. associate-*l/ 75.4%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 100.0%

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

6. Applied egg-rr 100.0%

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

   1. exp-1-e 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{1}{1 + {e}^{\left(b - a\right)}} \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		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) <= 0.8d0) 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) <= 0.8) {
		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) <= 0.8:
		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) <= 0.8)
		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) <= 0.8)
		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], 0.8], 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) < 0.80000000000000004

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-*l/ 98.4%

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

      3. associate-/r/ 98.4%

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

      4. remove-double-neg 98.4%

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

      5. unsub-neg 98.4%

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

      6. div-sub 3.1%

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

      7. *-lft-identity 3.1%

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

      8. associate-*l/ 3.1%

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

      9. lft-mult-inverse 98.4%

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

      10. sub-neg 98.4%

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

      11. distribute-frac-neg 98.4%

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

      12. remove-double-neg 98.4%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 99.5%

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

      1. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   if 0.80000000000000004 < (exp.f64 a)

   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%

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

      3. associate-/r/ 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 99.5%

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

      7. *-lft-identity 99.5%

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

      8. associate-*l/ 99.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 98.2%

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

      1. +-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-245}:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-245) (/ (exp a) b) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-245) {
		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) <= 1d-245) 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) <= 1e-245) {
		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) <= 1e-245:
		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) <= 1e-245)
		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) <= 1e-245)
		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], 1e-245], 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) < 9.9999999999999993e-246

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-*l/ 98.4%

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

      3. associate-/r/ 98.4%

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

      4. remove-double-neg 98.4%

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

      5. unsub-neg 98.4%

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

      6. div-sub 1.6%

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

      7. *-lft-identity 1.6%

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

      8. associate-*l/ 1.6%

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

      9. lft-mult-inverse 98.4%

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

      10. sub-neg 98.4%

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

      11. distribute-frac-neg 98.4%

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

      12. remove-double-neg 98.4%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 98.6%

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

   if 9.9999999999999993e-246 < (exp.f64 a)

   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%

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

      3. associate-/r/ 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 99.5%

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

      7. *-lft-identity 99.5%

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

      8. associate-*l/ 99.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 97.8%

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

      1. +-commutative 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-245}:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -360.0)
   (/ (exp a) b)
   (/
    1.0
    (+
     2.0
     (+ b (* a (+ (* a (- 0.5 (/ (* (* b 3.0) (* b 0.5)) b))) (- -1.0 b))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -360.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-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 (a <= (-360.0d0)) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * (0.5d0 - (((b * 3.0d0) * (b * 0.5d0)) / b))) + ((-1.0d0) - b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -360.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -360.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -360.0)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(0.5 - Float64(Float64(Float64(b * 3.0) * Float64(b * 0.5)) / b))) + Float64(-1.0 - b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -360.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -360.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(0.5 - N[(N[(N[(b * 3.0), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -360

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-*l/ 98.4%

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

      3. associate-/r/ 98.4%

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

      4. remove-double-neg 98.4%

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

      5. unsub-neg 98.4%

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

      6. div-sub 1.6%

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

      7. *-lft-identity 1.6%

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

      8. associate-*l/ 1.6%

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

      9. lft-mult-inverse 98.4%

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

      10. sub-neg 98.4%

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

      11. distribute-frac-neg 98.4%

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

      12. remove-double-neg 98.4%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 98.6%

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

   if -360 < a

   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%

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

      3. associate-/r/ 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 99.5%

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

      7. *-lft-identity 99.5%

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

      8. associate-*l/ 99.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 52.5%

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

      1. +-commutative 52.5%

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

      2. distribute-rgt1-in 52.5%

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

      3. rec-exp 52.5%

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

      4. associate-*r/ 52.5%

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

      5. *-rgt-identity 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in a around 0 51.6%

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

      1. flip-+ 50.1%

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

      2. mul-1-neg 50.1%

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

      3. mul-1-neg 50.1%

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

      4. sqr-neg 50.1%

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

      5. div-sub 50.1%

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

      6. pow2 50.1%

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

      7. add-sqr-sqrt 19.8%

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

      8. sqrt-unprod 50.1%

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

      9. mul-1-neg 50.1%

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

      10. mul-1-neg 50.1%

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

      11. sqr-neg 50.1%

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

      12. sqrt-unprod 45.3%

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

      13. add-sqr-sqrt 65.5%

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

      14. pow2 65.5%

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

   10. Applied egg-rr 50.1%

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

       1. div-sub 50.1%

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

       2. unpow2 50.1%

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

       3. unpow2 50.1%

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

       4. difference-of-squares 65.5%

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

       5. *-lft-identity 65.5%

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

       6. distribute-rgt-out 65.5%

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

       7. metadata-eval 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{1.5}\right) \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       8. cancel-sign-sub-inv 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(b + \left(-0.5\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(b + \color{blue}{-0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       10. distribute-rgt1-in 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(\left(-0.5 + 1\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       11. metadata-eval 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(\color{blue}{0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       12. cancel-sign-sub-inv 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{b + \left(-0.5\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       13. metadata-eval 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{b + \color{blue}{-0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       14. distribute-rgt1-in 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{\left(-0.5 + 1\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       15. metadata-eval 65.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 65.5%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]
   13. Step-by-step derivation

       1. times-frac 51.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{0.5 \cdot b}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. *-commutative 51.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{b \cdot 1.5}{0.5} \cdot \frac{\color{blue}{b \cdot 0.5}}{b}\right) - \left(1 + b\right)\right)\right)} \]

   14. Applied egg-rr 51.6%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{b \cdot 0.5}{b}}\right) - \left(1 + b\right)\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r/ 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\frac{b \cdot 1.5}{0.5} \cdot \left(b \cdot 0.5\right)}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. associate-/l* 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(b \cdot \frac{1.5}{0.5}\right)} \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

       3. metadata-eval 65.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{3}\right) \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

   16. Simplified 65.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 75.4%

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

   7. *-lft-identity 75.4%

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

   8. associate-*l/ 75.4%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + e^{b - a}} \]
6. Add Preprocessing

Alternative 6: 68.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(b \cdot 0.5\right) \cdot \left(a + \frac{a}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.5e-190)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (if (<= b 3e-9)
     (/ 1.0 (+ 2.0 (+ b (* a (+ (* (* b 0.5) (+ a (/ a b))) (- -1.0 b))))))
     (/
      1.0
      (+
       2.0
       (+
        b
        (* a (+ (* a (- 0.5 (/ (* (* b 3.0) (* b 0.5)) b))) (- -1.0 b)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.5e-190) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 3e-9) {
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-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 <= (-4.5d-190)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (b <= 3d-9) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * (((b * 0.5d0) * (a + (a / b))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * (0.5d0 - (((b * 3.0d0) * (b * 0.5d0)) / b))) + ((-1.0d0) - b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.5e-190) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 3e-9) {
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.5e-190:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	elif b <= 3e-9:
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.5e-190)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	elseif (b <= 3e-9)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(Float64(b * 0.5) * Float64(a + Float64(a / b))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(0.5 - Float64(Float64(Float64(b * 3.0) * Float64(b * 0.5)) / b))) + Float64(-1.0 - b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.5e-190)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	elseif (b <= 3e-9)
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.5e-190], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-9], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(N[(b * 0.5), $MachinePrecision] * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(0.5 - N[(N[(N[(b * 3.0), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000021e-190

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. remove-double-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. div-sub 83.8%

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

      7. *-lft-identity 83.8%

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

      8. associate-*l/ 83.8%

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

      9. lft-mult-inverse 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-frac-neg 98.8%

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

      12. remove-double-neg 98.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 54.0%

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

      1. +-commutative 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in a around 0 47.3%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if -4.50000000000000021e-190 < b < 2.99999999999999998e-9

   1. Initial program 99.0%

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

      1. *-lft-identity 99.0%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

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

      6. div-sub 69.2%

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

      7. *-lft-identity 69.2%

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

      8. associate-*l/ 69.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 84.7%

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

   9. Taylor expanded in b around inf 93.6%

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

       1. distribute-lft-out 93.6%

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

       2. associate-*r* 93.6%

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

       3. *-commutative 93.6%

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

   11. Simplified 93.6%

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

   if 2.99999999999999998e-9 < 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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 75.4%

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

      7. *-lft-identity 75.4%

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

      8. associate-*l/ 75.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 30.1%

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

      1. +-commutative 30.1%

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

      2. distribute-rgt1-in 30.1%

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

      3. rec-exp 30.1%

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

      4. associate-*r/ 30.1%

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

      5. *-rgt-identity 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in a around 0 26.0%

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

      1. flip-+ 11.2%

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

      2. mul-1-neg 11.2%

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

      3. mul-1-neg 11.2%

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

      4. sqr-neg 11.2%

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

      5. div-sub 11.2%

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

      6. pow2 11.2%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 11.2%

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

      9. mul-1-neg 11.2%

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

      10. mul-1-neg 11.2%

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

      11. sqr-neg 11.2%

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

      12. sqrt-unprod 64.8%

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

      13. add-sqr-sqrt 64.8%

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

      14. pow2 64.8%

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

   10. Applied egg-rr 11.2%

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

       1. div-sub 11.2%

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

       2. unpow2 11.2%

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

       3. unpow2 11.2%

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

       4. difference-of-squares 64.8%

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

       5. *-lft-identity 64.8%

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

       6. distribute-rgt-out 64.8%

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

       7. metadata-eval 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{1.5}\right) \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       8. cancel-sign-sub-inv 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(b + \left(-0.5\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(b + \color{blue}{-0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       10. distribute-rgt1-in 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(\left(-0.5 + 1\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       11. metadata-eval 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(\color{blue}{0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       12. cancel-sign-sub-inv 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{b + \left(-0.5\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       13. metadata-eval 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{b + \color{blue}{-0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       14. distribute-rgt1-in 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{\left(-0.5 + 1\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       15. metadata-eval 64.8%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 64.8%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]
   13. Step-by-step derivation

       1. times-frac 26.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{0.5 \cdot b}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. *-commutative 26.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{b \cdot 1.5}{0.5} \cdot \frac{\color{blue}{b \cdot 0.5}}{b}\right) - \left(1 + b\right)\right)\right)} \]

   14. Applied egg-rr 26.0%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{b \cdot 0.5}{b}}\right) - \left(1 + b\right)\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r/ 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\frac{b \cdot 1.5}{0.5} \cdot \left(b \cdot 0.5\right)}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. associate-/l* 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(b \cdot \frac{1.5}{0.5}\right)} \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

       3. metadata-eval 64.8%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{3}\right) \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

   16. Simplified 64.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(b \cdot 0.5\right) \cdot \left(a + \frac{a}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 0.5 + -1\right)\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 10^{-179}:\\ \;\;\;\;\frac{1}{2 + \left(b + b \cdot \left(t\_0 + \frac{t\_0}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ (* a 0.5) -1.0))))
   (if (<= b -5.8e-282)
     (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
     (if (<= b 1e-179)
       (/ 1.0 (+ 2.0 (+ b (* b (+ t_0 (/ t_0 b))))))
       (/
        1.0
        (+
         2.0
         (+
          b
          (*
           a
           (+ (* a (- 0.5 (/ (* (* b 3.0) (* b 0.5)) b))) (- -1.0 b))))))))))

C

double code(double a, double b) {
	double t_0 = a * ((a * 0.5) + -1.0);
	double tmp;
	if (b <= -5.8e-282) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1e-179) {
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + (t_0 / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * ((a * 0.5d0) + (-1.0d0))
    if (b <= (-5.8d-282)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (b <= 1d-179) then
        tmp = 1.0d0 / (2.0d0 + (b + (b * (t_0 + (t_0 / b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * (0.5d0 - (((b * 3.0d0) * (b * 0.5d0)) / b))) + ((-1.0d0) - b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * ((a * 0.5) + -1.0);
	double tmp;
	if (b <= -5.8e-282) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1e-179) {
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + (t_0 / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * ((a * 0.5) + -1.0)
	tmp = 0
	if b <= -5.8e-282:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	elif b <= 1e-179:
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + (t_0 / b)))))
	else:
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(Float64(a * 0.5) + -1.0))
	tmp = 0.0
	if (b <= -5.8e-282)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	elseif (b <= 1e-179)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(b * Float64(t_0 + Float64(t_0 / b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(0.5 - Float64(Float64(Float64(b * 3.0) * Float64(b * 0.5)) / b))) + Float64(-1.0 - b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * ((a * 0.5) + -1.0);
	tmp = 0.0;
	if (b <= -5.8e-282)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	elseif (b <= 1e-179)
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + (t_0 / b)))));
	else
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (((b * 3.0) * (b * 0.5)) / b))) + (-1.0 - b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.8e-282], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-179], N[(1.0 / N[(2.0 + N[(b + N[(b * N[(t$95$0 + N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(0.5 - N[(N[(N[(b * 3.0), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.79999999999999995e-282

   1. Initial program 99.0%

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

      1. *-lft-identity 99.0%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

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

      6. div-sub 80.0%

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

      7. *-lft-identity 80.0%

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

      8. associate-*l/ 80.0%

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

      9. lft-mult-inverse 99.0%

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

      10. sub-neg 99.0%

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

      11. distribute-frac-neg 99.0%

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

      12. remove-double-neg 99.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 63.2%

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

      1. +-commutative 63.2%

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

   7. Simplified 63.2%

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

   8. Taylor expanded in a around 0 56.0%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if -5.79999999999999995e-282 < b < 1e-179

   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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 56.1%

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

      7. *-lft-identity 56.1%

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

      8. associate-*l/ 56.1%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 90.2%

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

      1. +-commutative 90.2%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 74.5%

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

   9. Taylor expanded in b around inf 97.7%

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

   if 1e-179 < b

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*l/ 99.1%

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

      3. associate-/r/ 99.1%

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

      4. remove-double-neg 99.1%

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

      5. unsub-neg 99.1%

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

      6. div-sub 78.3%

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

      7. *-lft-identity 78.3%

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

      8. associate-*l/ 78.3%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 58.1%

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

      1. +-commutative 58.1%

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

      2. distribute-rgt1-in 58.1%

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

      3. rec-exp 58.1%

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

      4. associate-*r/ 58.1%

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

      5. *-rgt-identity 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in a around 0 52.2%

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

      1. flip-+ 43.3%

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

      2. mul-1-neg 43.3%

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

      3. mul-1-neg 43.3%

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

      4. sqr-neg 43.3%

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

      5. div-sub 43.3%

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

      6. pow2 43.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 43.3%

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

      9. mul-1-neg 43.3%

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

      10. mul-1-neg 43.3%

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

      11. sqr-neg 43.3%

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

      12. sqrt-unprod 75.5%

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

      13. add-sqr-sqrt 75.5%

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

      14. pow2 75.5%

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

   10. Applied egg-rr 43.3%

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

       1. div-sub 43.3%

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

       2. unpow2 43.3%

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

       3. unpow2 43.3%

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

       4. difference-of-squares 75.5%

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

       5. *-lft-identity 75.5%

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

       6. distribute-rgt-out 75.5%

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

       7. metadata-eval 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{1.5}\right) \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       8. cancel-sign-sub-inv 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(b + \left(-0.5\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(b + \color{blue}{-0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       10. distribute-rgt1-in 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(\left(-0.5 + 1\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       11. metadata-eval 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(\color{blue}{0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       12. cancel-sign-sub-inv 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{b + \left(-0.5\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       13. metadata-eval 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{b + \color{blue}{-0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       14. distribute-rgt1-in 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{\left(-0.5 + 1\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       15. metadata-eval 75.5%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 75.5%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]
   13. Step-by-step derivation

       1. times-frac 52.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{0.5 \cdot b}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. *-commutative 52.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{b \cdot 1.5}{0.5} \cdot \frac{\color{blue}{b \cdot 0.5}}{b}\right) - \left(1 + b\right)\right)\right)} \]

   14. Applied egg-rr 52.2%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{b \cdot 1.5}{0.5} \cdot \frac{b \cdot 0.5}{b}}\right) - \left(1 + b\right)\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r/ 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\frac{b \cdot 1.5}{0.5} \cdot \left(b \cdot 0.5\right)}{b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. associate-/l* 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(b \cdot \frac{1.5}{0.5}\right)} \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

       3. metadata-eval 75.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{3}\right) \cdot \left(b \cdot 0.5\right)}{b}\right) - \left(1 + b\right)\right)\right)} \]

   16. Simplified 75.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 10^{-179}:\\ \;\;\;\;\frac{1}{2 + \left(b + b \cdot \left(a \cdot \left(a \cdot 0.5 + -1\right) + \frac{a \cdot \left(a \cdot 0.5 + -1\right)}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 3\right) \cdot \left(b \cdot 0.5\right)}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - b \cdot 1.5\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.15e-211)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (if (<= b 1.6e-237)
     (/ 1.0 (* b (+ (- 1.0 a) (/ (- 2.0 a) b))))
     (if (<= b 5e+78)
       (/ 1.0 (+ 2.0 (+ b (* a (+ (* a (- 0.5 (* b 1.5))) (- -1.0 b))))))
       (/ 1.0 (* a (- -1.0 (- b (/ (+ b 2.0) a)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.15e-211) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1.6e-237) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 5e+78) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (b * 1.5))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / 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 <= (-2.15d-211)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (b <= 1.6d-237) then
        tmp = 1.0d0 / (b * ((1.0d0 - a) + ((2.0d0 - a) / b)))
    else if (b <= 5d+78) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * (0.5d0 - (b * 1.5d0))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) - (b - ((b + 2.0d0) / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.15e-211) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1.6e-237) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 5e+78) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (b * 1.5))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.15e-211:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	elif b <= 1.6e-237:
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)))
	elif b <= 5e+78:
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (b * 1.5))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.15e-211)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	elseif (b <= 1.6e-237)
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.0 - a) + Float64(Float64(2.0 - a) / b))));
	elseif (b <= 5e+78)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(0.5 - Float64(b * 1.5))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - Float64(b - Float64(Float64(b + 2.0) / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.15e-211)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	elseif (b <= 1.6e-237)
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	elseif (b <= 5e+78)
		tmp = 1.0 / (2.0 + (b + (a * ((a * (0.5 - (b * 1.5))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.15e-211], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-237], N[(1.0 / N[(b * N[(N[(1.0 - a), $MachinePrecision] + N[(N[(2.0 - a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+78], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(0.5 - N[(b * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 - N[(b - N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.15e-211

   1. Initial program 98.8%

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

      1. *-lft-identity 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. remove-double-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. div-sub 83.7%

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

      7. *-lft-identity 83.7%

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

      8. associate-*l/ 83.7%

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

      9. lft-mult-inverse 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-frac-neg 98.8%

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

      12. remove-double-neg 98.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in a around 0 51.0%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if -2.15e-211 < b < 1.6e-237

   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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 56.8%

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

      7. *-lft-identity 56.8%

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

      8. associate-*l/ 56.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 57.9%

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

      1. associate-+r+ 57.9%

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

      2. associate-*r* 57.9%

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

      3. distribute-rgt1-in 57.9%

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

      4. +-commutative 57.9%

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

      5. neg-mul-1 57.9%

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

      6. unsub-neg 57.9%

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

   10. Simplified 57.9%

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

   11. Taylor expanded in b around -inf 96.5%

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

       1. mul-1-neg 96.5%

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

       2. *-commutative 96.5%

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

       3. distribute-rgt-neg-in 96.5%

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

       4. distribute-lft-out 96.5%

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

       5. mul-1-neg 96.5%

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

   13. Simplified 96.5%

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

   if 1.6e-237 < b < 4.99999999999999984e78

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 74.4%

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

      7. *-lft-identity 74.4%

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

      8. associate-*l/ 74.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. distribute-rgt1-in 83.5%

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

      3. rec-exp 83.5%

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

      4. associate-*r/ 83.5%

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

      5. *-rgt-identity 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in a around 0 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. fma-define 73.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 73.6%

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

      5. mul-1-neg 73.6%

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

      6. mul-1-neg 73.6%

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

      7. sqr-neg 73.6%

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

      8. sqrt-unprod 73.6%

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

      9. add-sqr-sqrt 73.6%

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

   10. Applied egg-rr 73.6%

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

       1. fma-undefine 73.6%

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

       2. distribute-rgt-out 73.6%

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

       3. metadata-eval 73.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - b \cdot \color{blue}{1.5}\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 73.6%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{b \cdot 1.5}\right) - \left(1 + b\right)\right)\right)} \]

   if 4.99999999999999984e78 < 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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 79.2%

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

      7. *-lft-identity 79.2%

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

      8. associate-*l/ 79.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 26.3%

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

      1. +-commutative 26.3%

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

      2. distribute-rgt1-in 26.3%

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

      3. rec-exp 26.3%

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

      4. associate-*r/ 26.3%

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

      5. *-rgt-identity 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in a around 0 22.4%

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

      1. associate-+r+ 22.4%

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

      2. associate-*r* 22.4%

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

      3. distribute-rgt1-in 22.4%

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

      4. +-commutative 22.4%

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

      5. neg-mul-1 22.4%

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

      6. unsub-neg 22.4%

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

   10. Simplified 22.4%

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

   11. Taylor expanded in a around -inf 66.4%

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

       1. mul-1-neg 66.4%

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

       2. distribute-rgt-neg-in 66.4%

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

       3. mul-1-neg 66.4%

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

       4. unsub-neg 66.4%

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

       5. +-commutative 66.4%

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

   13. Simplified 66.4%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b - \frac{b + 2}{a}\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - b \cdot 1.5\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(b \cdot 0.5\right) \cdot \left(a + \frac{a}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.2e-192)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (if (<= b 7.6e+76)
     (/ 1.0 (+ 2.0 (+ b (* a (+ (* (* b 0.5) (+ a (/ a b))) (- -1.0 b))))))
     (/ 1.0 (* a (- -1.0 (- b (/ (+ b 2.0) a))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -5.2e-192) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / 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 <= (-5.2d-192)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (b <= 7.6d+76) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * (((b * 0.5d0) * (a + (a / b))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) - (b - ((b + 2.0d0) / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -5.2e-192) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -5.2e-192:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	elif b <= 7.6e+76:
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -5.2e-192)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	elseif (b <= 7.6e+76)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(Float64(b * 0.5) * Float64(a + Float64(a / b))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - Float64(b - Float64(Float64(b + 2.0) / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -5.2e-192)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	elseif (b <= 7.6e+76)
		tmp = 1.0 / (2.0 + (b + (a * (((b * 0.5) * (a + (a / b))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -5.2e-192], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e+76], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(N[(b * 0.5), $MachinePrecision] * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 - N[(b - N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000003e-192

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. remove-double-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. div-sub 83.8%

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

      7. *-lft-identity 83.8%

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

      8. associate-*l/ 83.8%

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

      9. lft-mult-inverse 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-frac-neg 98.8%

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

      12. remove-double-neg 98.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 54.0%

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

      1. +-commutative 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in a around 0 47.3%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if -5.2000000000000003e-192 < b < 7.60000000000000049e76

   1. Initial program 99.2%

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

      1. *-lft-identity 99.2%

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

      2. associate-*l/ 99.2%

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

      3. associate-/r/ 99.2%

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

      4. remove-double-neg 99.2%

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

      5. unsub-neg 99.2%

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

      6. div-sub 68.8%

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

      7. *-lft-identity 68.8%

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

      8. associate-*l/ 68.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 81.4%

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

      1. +-commutative 81.4%

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

      2. distribute-rgt1-in 89.9%

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

      3. rec-exp 90.0%

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

      4. associate-*r/ 90.0%

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

      5. *-rgt-identity 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in a around 0 75.7%

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

   9. Taylor expanded in b around inf 83.1%

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

       1. distribute-lft-out 83.1%

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

       2. associate-*r* 83.1%

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

       3. *-commutative 83.1%

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

   11. Simplified 83.1%

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

   if 7.60000000000000049e76 < 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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 79.2%

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

      7. *-lft-identity 79.2%

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

      8. associate-*l/ 79.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 26.3%

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

      1. +-commutative 26.3%

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

      2. distribute-rgt1-in 26.3%

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

      3. rec-exp 26.3%

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

      4. associate-*r/ 26.3%

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

      5. *-rgt-identity 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in a around 0 22.4%

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

      1. associate-+r+ 22.4%

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

      2. associate-*r* 22.4%

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

      3. distribute-rgt1-in 22.4%

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

      4. +-commutative 22.4%

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

      5. neg-mul-1 22.4%

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

      6. unsub-neg 22.4%

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

   10. Simplified 22.4%

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

   11. Taylor expanded in a around -inf 66.4%

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

       1. mul-1-neg 66.4%

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

       2. distribute-rgt-neg-in 66.4%

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

       3. mul-1-neg 66.4%

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

       4. unsub-neg 66.4%

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

       5. +-commutative 66.4%

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

   13. Simplified 66.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(b \cdot 0.5\right) \cdot \left(a + \frac{a}{b}\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 0.5 + -1\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{1}{2 + t\_0}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ (* a 0.5) -1.0))))
   (if (<= b -2e-210)
     (/ 1.0 (+ 2.0 t_0))
     (if (<= b 2.4e-239)
       (/ 1.0 (* b (+ (- 1.0 a) (/ (- 2.0 a) b))))
       (if (<= b 7.6e+76)
         (/ 1.0 (+ 2.0 (+ b t_0)))
         (/ 1.0 (* a (- -1.0 (- b (/ (+ b 2.0) a))))))))))

C

double code(double a, double b) {
	double t_0 = a * ((a * 0.5) + -1.0);
	double tmp;
	if (b <= -2e-210) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 2.4e-239) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	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 = a * ((a * 0.5d0) + (-1.0d0))
    if (b <= (-2d-210)) then
        tmp = 1.0d0 / (2.0d0 + t_0)
    else if (b <= 2.4d-239) then
        tmp = 1.0d0 / (b * ((1.0d0 - a) + ((2.0d0 - a) / b)))
    else if (b <= 7.6d+76) then
        tmp = 1.0d0 / (2.0d0 + (b + t_0))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) - (b - ((b + 2.0d0) / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * ((a * 0.5) + -1.0);
	double tmp;
	if (b <= -2e-210) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 2.4e-239) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * ((a * 0.5) + -1.0)
	tmp = 0
	if b <= -2e-210:
		tmp = 1.0 / (2.0 + t_0)
	elif b <= 2.4e-239:
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)))
	elif b <= 7.6e+76:
		tmp = 1.0 / (2.0 + (b + t_0))
	else:
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(Float64(a * 0.5) + -1.0))
	tmp = 0.0
	if (b <= -2e-210)
		tmp = Float64(1.0 / Float64(2.0 + t_0));
	elseif (b <= 2.4e-239)
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.0 - a) + Float64(Float64(2.0 - a) / b))));
	elseif (b <= 7.6e+76)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + t_0)));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - Float64(b - Float64(Float64(b + 2.0) / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * ((a * 0.5) + -1.0);
	tmp = 0.0;
	if (b <= -2e-210)
		tmp = 1.0 / (2.0 + t_0);
	elseif (b <= 2.4e-239)
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	elseif (b <= 7.6e+76)
		tmp = 1.0 / (2.0 + (b + t_0));
	else
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-210], N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-239], N[(1.0 / N[(b * N[(N[(1.0 - a), $MachinePrecision] + N[(N[(2.0 - a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e+76], N[(1.0 / N[(2.0 + N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 - N[(b - N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0000000000000001e-210

   1. Initial program 98.8%

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

      1. *-lft-identity 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. remove-double-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. div-sub 83.7%

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

      7. *-lft-identity 83.7%

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

      8. associate-*l/ 83.7%

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

      9. lft-mult-inverse 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-frac-neg 98.8%

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

      12. remove-double-neg 98.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in a around 0 46.5%

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

   if -2.0000000000000001e-210 < b < 2.39999999999999993e-239

   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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 56.8%

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

      7. *-lft-identity 56.8%

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

      8. associate-*l/ 56.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 57.9%

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

      1. associate-+r+ 57.9%

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

      2. associate-*r* 57.9%

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

      3. distribute-rgt1-in 57.9%

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

      4. +-commutative 57.9%

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

      5. neg-mul-1 57.9%

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

      6. unsub-neg 57.9%

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

   10. Simplified 57.9%

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

   11. Taylor expanded in b around -inf 96.5%

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

       1. mul-1-neg 96.5%

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

       2. *-commutative 96.5%

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

       3. distribute-rgt-neg-in 96.5%

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

       4. distribute-lft-out 96.5%

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

       5. mul-1-neg 96.5%

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

   13. Simplified 96.5%

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

   if 2.39999999999999993e-239 < b < 7.60000000000000049e76

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 74.4%

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

      7. *-lft-identity 74.4%

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

      8. associate-*l/ 74.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. distribute-rgt1-in 83.5%

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

      3. rec-exp 83.5%

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

      4. associate-*r/ 83.5%

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

      5. *-rgt-identity 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in a around 0 73.6%

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

   9. Taylor expanded in b around inf 73.6%

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

       1. distribute-lft-out 73.6%

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

       2. associate-*r* 73.6%

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

       3. *-commutative 73.6%

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

   11. Simplified 73.6%

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

   12. Taylor expanded in b around 0 73.6%

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

   if 7.60000000000000049e76 < 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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 79.2%

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

      7. *-lft-identity 79.2%

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

      8. associate-*l/ 79.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 26.3%

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

      1. +-commutative 26.3%

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

      2. distribute-rgt1-in 26.3%

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

      3. rec-exp 26.3%

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

      4. associate-*r/ 26.3%

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

      5. *-rgt-identity 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in a around 0 22.4%

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

      1. associate-+r+ 22.4%

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

      2. associate-*r* 22.4%

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

      3. distribute-rgt1-in 22.4%

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

      4. +-commutative 22.4%

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

      5. neg-mul-1 22.4%

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

      6. unsub-neg 22.4%

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

   10. Simplified 22.4%

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

   11. Taylor expanded in a around -inf 66.4%

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

       1. mul-1-neg 66.4%

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

       2. distribute-rgt-neg-in 66.4%

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

       3. mul-1-neg 66.4%

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

       4. unsub-neg 66.4%

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

       5. +-commutative 66.4%

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

   13. Simplified 66.4%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b - \frac{b + 2}{a}\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-210}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot 0.5 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot 0.5 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.5e-211)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (if (<= b 1.1e-237)
     (/ 1.0 (* b (+ (- 1.0 a) (/ (- 2.0 a) b))))
     (if (<= b 7.6e+76)
       (/ 1.0 (+ 2.0 (+ b (* a (+ (* a 0.5) -1.0)))))
       (/ 1.0 (* a (- -1.0 (- b (/ (+ b 2.0) a)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.5e-211) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1.1e-237) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * 0.5) + -1.0))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / 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 <= (-1.5d-211)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (b <= 1.1d-237) then
        tmp = 1.0d0 / (b * ((1.0d0 - a) + ((2.0d0 - a) / b)))
    else if (b <= 7.6d+76) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * 0.5d0) + (-1.0d0)))))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) - (b - ((b + 2.0d0) / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.5e-211) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else if (b <= 1.1e-237) {
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	} else if (b <= 7.6e+76) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * 0.5) + -1.0))));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.5e-211:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	elif b <= 1.1e-237:
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)))
	elif b <= 7.6e+76:
		tmp = 1.0 / (2.0 + (b + (a * ((a * 0.5) + -1.0))))
	else:
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.5e-211)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	elseif (b <= 1.1e-237)
		tmp = Float64(1.0 / Float64(b * Float64(Float64(1.0 - a) + Float64(Float64(2.0 - a) / b))));
	elseif (b <= 7.6e+76)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * 0.5) + -1.0)))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - Float64(b - Float64(Float64(b + 2.0) / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.5e-211)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	elseif (b <= 1.1e-237)
		tmp = 1.0 / (b * ((1.0 - a) + ((2.0 - a) / b)));
	elseif (b <= 7.6e+76)
		tmp = 1.0 / (2.0 + (b + (a * ((a * 0.5) + -1.0))));
	else
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.5e-211], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-237], N[(1.0 / N[(b * N[(N[(1.0 - a), $MachinePrecision] + N[(N[(2.0 - a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e+76], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 - N[(b - N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.50000000000000002e-211

   1. Initial program 98.8%

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

      1. *-lft-identity 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. remove-double-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. div-sub 83.7%

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

      7. *-lft-identity 83.7%

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

      8. associate-*l/ 83.7%

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

      9. lft-mult-inverse 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-frac-neg 98.8%

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

      12. remove-double-neg 98.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in a around 0 51.0%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if -1.50000000000000002e-211 < b < 1.09999999999999999e-237

   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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 56.8%

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

      7. *-lft-identity 56.8%

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

      8. associate-*l/ 56.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. distribute-rgt1-in 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 57.9%

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

      1. associate-+r+ 57.9%

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

      2. associate-*r* 57.9%

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

      3. distribute-rgt1-in 57.9%

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

      4. +-commutative 57.9%

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

      5. neg-mul-1 57.9%

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

      6. unsub-neg 57.9%

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

   10. Simplified 57.9%

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

   11. Taylor expanded in b around -inf 96.5%

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

       1. mul-1-neg 96.5%

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

       2. *-commutative 96.5%

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

       3. distribute-rgt-neg-in 96.5%

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

       4. distribute-lft-out 96.5%

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

       5. mul-1-neg 96.5%

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

   13. Simplified 96.5%

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

   if 1.09999999999999999e-237 < b < 7.60000000000000049e76

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 74.4%

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

      7. *-lft-identity 74.4%

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

      8. associate-*l/ 74.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. distribute-rgt1-in 83.5%

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

      3. rec-exp 83.5%

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

      4. associate-*r/ 83.5%

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

      5. *-rgt-identity 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in a around 0 73.6%

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

   9. Taylor expanded in b around inf 73.6%

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

       1. distribute-lft-out 73.6%

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

       2. associate-*r* 73.6%

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

       3. *-commutative 73.6%

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

   11. Simplified 73.6%

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

   12. Taylor expanded in b around 0 73.6%

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

   if 7.60000000000000049e76 < 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%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 79.2%

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

      7. *-lft-identity 79.2%

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

      8. associate-*l/ 79.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 26.3%

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

      1. +-commutative 26.3%

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

      2. distribute-rgt1-in 26.3%

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

      3. rec-exp 26.3%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 26.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 26.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 26.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 22.4%

      \[\leadsto \frac{1}{\color{blue}{\left(1 + \left(b + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)} + 1} \]
   9. Step-by-step derivation

      1. associate-+r+ 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + b\right) + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)} + 1} \]

      2. associate-*r* 22.4%

         \[\leadsto \frac{1}{\left(\left(1 + b\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(1 + b\right)}\right) + 1} \]

      3. distribute-rgt1-in 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot a + 1\right) \cdot \left(1 + b\right)} + 1} \]

      4. +-commutative 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 + -1 \cdot a\right)} \cdot \left(1 + b\right) + 1} \]

      5. neg-mul-1 22.4%

         \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-a\right)}\right) \cdot \left(1 + b\right) + 1} \]

      6. unsub-neg 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right)} \cdot \left(1 + b\right) + 1} \]

   10. Simplified 22.4%

       \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right) \cdot \left(1 + b\right)} + 1} \]

   11. Taylor expanded in a around -inf 66.4%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(a \cdot \left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. mul-1-neg 66.4%

          \[\leadsto \frac{1}{\color{blue}{-a \cdot \left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)}} \]

       2. distribute-rgt-neg-in 66.4%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)\right)}} \]

       3. mul-1-neg 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \left(b + \color{blue}{\left(-\frac{2 + b}{a}\right)}\right)\right)\right)} \]

       4. unsub-neg 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \color{blue}{\left(b - \frac{2 + b}{a}\right)}\right)\right)} \]

       5. +-commutative 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \left(b - \frac{\color{blue}{b + 2}}{a}\right)\right)\right)} \]

   13. Simplified 66.4%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b - \frac{b + 2}{a}\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{b \cdot \left(\left(1 - a\right) + \frac{2 - a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot 0.5 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + a \cdot \left(a \cdot -1.5 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.0)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (* b (+ 1.0 (* a (+ (* a -1.5) -1.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (b * (1.0 + (a * ((a * -1.5) + -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 <= 2.0d0) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / (b * (1.0d0 + (a * ((a * (-1.5d0)) + (-1.0d0)))))
    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 * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (b * (1.0 + (a * ((a * -1.5) + -1.0))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.0:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (b * (1.0 + (a * ((a * -1.5) + -1.0))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(b * Float64(1.0 + Float64(a * Float64(Float64(a * -1.5) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (b * (1.0 + (a * ((a * -1.5) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.0], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(1.0 + N[(a * N[(N[(a * -1.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 75.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 75.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 75.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 79.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 79.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 66.6%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 2 < 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%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 76.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 76.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 76.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 28.5%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 28.5%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 28.5%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 28.5%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 28.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 28.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 28.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 24.3%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. flip-+ 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\left(-1 \cdot b\right) \cdot \left(-1 \cdot b\right) - \left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

      2. mul-1-neg 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(-b\right)} \cdot \left(-1 \cdot b\right) - \left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

      3. mul-1-neg 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(-b\right) \cdot \color{blue}{\left(-b\right)} - \left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

      4. sqr-neg 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{b \cdot b} - \left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

      5. div-sub 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\left(\frac{b \cdot b}{-1 \cdot b - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)}\right) - \left(1 + b\right)\right)\right)} \]

      6. pow2 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{\color{blue}{{b}^{2}}}{-1 \cdot b - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\color{blue}{\sqrt{-1 \cdot b} \cdot \sqrt{-1 \cdot b}} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      8. sqrt-unprod 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\color{blue}{\sqrt{\left(-1 \cdot b\right) \cdot \left(-1 \cdot b\right)}} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      9. mul-1-neg 9.0%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\sqrt{\color{blue}{\left(-b\right)} \cdot \left(-1 \cdot b\right)} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      10. mul-1-neg 9.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\sqrt{\left(-b\right) \cdot \color{blue}{\left(-b\right)}} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      11. sqr-neg 9.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\sqrt{\color{blue}{b \cdot b}} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      12. sqrt-unprod 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      13. add-sqr-sqrt 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{\color{blue}{b} - 0.5 \cdot b} - \frac{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

      14. pow2 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(\frac{{b}^{2}}{b - 0.5 \cdot b} - \frac{\color{blue}{{\left(0.5 \cdot b\right)}^{2}}}{-1 \cdot b - 0.5 \cdot b}\right)\right) - \left(1 + b\right)\right)\right)} \]

   10. Applied egg-rr 9.0%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\left(\frac{{b}^{2}}{b - 0.5 \cdot b} - \frac{{\left(0.5 \cdot b\right)}^{2}}{b - 0.5 \cdot b}\right)}\right) - \left(1 + b\right)\right)\right)} \]
   11. Step-by-step derivation

       1. div-sub 9.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{{b}^{2} - {\left(0.5 \cdot b\right)}^{2}}{b - 0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       2. unpow2 9.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{b \cdot b} - {\left(0.5 \cdot b\right)}^{2}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       3. unpow2 9.0%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{b \cdot b - \color{blue}{\left(0.5 \cdot b\right) \cdot \left(0.5 \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       4. difference-of-squares 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(b + 0.5 \cdot b\right) \cdot \left(b - 0.5 \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       5. *-lft-identity 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(\color{blue}{1 \cdot b} + 0.5 \cdot b\right) \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       6. distribute-rgt-out 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\color{blue}{\left(b \cdot \left(1 + 0.5\right)\right)} \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       7. metadata-eval 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot \color{blue}{1.5}\right) \cdot \left(b - 0.5 \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       8. cancel-sign-sub-inv 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(b + \left(-0.5\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 64.2%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(b + \color{blue}{-0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       10. distribute-rgt1-in 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \color{blue}{\left(\left(-0.5 + 1\right) \cdot b\right)}}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       11. metadata-eval 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(\color{blue}{0.5} \cdot b\right)}{b - 0.5 \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       12. cancel-sign-sub-inv 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{b + \left(-0.5\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       13. metadata-eval 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{b + \color{blue}{-0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

       14. distribute-rgt1-in 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{\left(-0.5 + 1\right) \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

       15. metadata-eval 64.2%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{\color{blue}{0.5} \cdot b}\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 64.2%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \color{blue}{\frac{\left(b \cdot 1.5\right) \cdot \left(0.5 \cdot b\right)}{0.5 \cdot b}}\right) - \left(1 + b\right)\right)\right)} \]

   13. Taylor expanded in b around inf 24.3%

       \[\leadsto \color{blue}{\frac{1}{b \cdot \left(1 + a \cdot \left(-1.5 \cdot a - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + a \cdot \left(a \cdot -1.5 + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.5% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(b \cdot \left(a \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e-23)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (+ 2.0 (+ b (* a (* b (* a 0.5))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e-23) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * (b * (a * 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 <= 5d-23) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b + (a * (b * (a * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e-23) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b + (a * (b * (a * 0.5)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e-23:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b + (a * (b * (a * 0.5)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e-23)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(b * Float64(a * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e-23)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b + (a * (b * (a * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e-23], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(b * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000002e-23

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.4%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.4%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 76.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 76.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 76.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 79.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 79.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 66.9%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 5.0000000000000002e-23 < b

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.6%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.6%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.6%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 98.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 74.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 74.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 74.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.8%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 34.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 34.0%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 34.0%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 34.0%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 34.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 34.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 34.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 27.5%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)}} \]

   9. Taylor expanded in b around inf 27.6%

      \[\leadsto \frac{1}{2 + \left(b + \color{blue}{a \cdot \left(b \cdot \left(0.5 \cdot a - 1\right)\right)}\right)} \]

   10. Taylor expanded in a around inf 27.6%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot b\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 27.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot 0.5\right)}\right)} \]

       2. associate-*r* 27.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(a \cdot \left(b \cdot 0.5\right)\right)}\right)} \]

       3. *-commutative 27.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(\left(b \cdot 0.5\right) \cdot a\right)}\right)} \]

       4. associate-*l* 27.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(b \cdot \left(0.5 \cdot a\right)\right)}\right)} \]

       5. *-commutative 27.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(b \cdot \color{blue}{\left(a \cdot 0.5\right)}\right)\right)} \]

   12. Simplified 27.6%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(b \cdot \left(a \cdot 0.5\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(b \cdot \left(a \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e+79)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (* a (- -1.0 (- b (/ (+ b 2.0) a)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e+79) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / 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 <= 4d+79) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) - (b - ((b + 2.0d0) / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4e+79) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4e+79:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e+79)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - Float64(b - Float64(Float64(b + 2.0) / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4e+79)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (a * (-1.0 - (b - ((b + 2.0) / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e+79], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 - N[(b - N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999987e79

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 74.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 74.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 74.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 75.7%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 75.7%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 62.7%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 3.99999999999999987e79 < 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%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 26.3%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 26.3%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 26.3%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 26.3%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 26.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 26.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 26.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 22.4%

      \[\leadsto \frac{1}{\color{blue}{\left(1 + \left(b + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)} + 1} \]
   9. Step-by-step derivation

      1. associate-+r+ 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + b\right) + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)} + 1} \]

      2. associate-*r* 22.4%

         \[\leadsto \frac{1}{\left(\left(1 + b\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(1 + b\right)}\right) + 1} \]

      3. distribute-rgt1-in 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot a + 1\right) \cdot \left(1 + b\right)} + 1} \]

      4. +-commutative 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 + -1 \cdot a\right)} \cdot \left(1 + b\right) + 1} \]

      5. neg-mul-1 22.4%

         \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-a\right)}\right) \cdot \left(1 + b\right) + 1} \]

      6. unsub-neg 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right)} \cdot \left(1 + b\right) + 1} \]

   10. Simplified 22.4%

       \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right) \cdot \left(1 + b\right)} + 1} \]

   11. Taylor expanded in a around -inf 66.4%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(a \cdot \left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. mul-1-neg 66.4%

          \[\leadsto \frac{1}{\color{blue}{-a \cdot \left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)}} \]

       2. distribute-rgt-neg-in 66.4%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b + -1 \cdot \frac{2 + b}{a}\right)\right)\right)}} \]

       3. mul-1-neg 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \left(b + \color{blue}{\left(-\frac{2 + b}{a}\right)}\right)\right)\right)} \]

       4. unsub-neg 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \color{blue}{\left(b - \frac{2 + b}{a}\right)}\right)\right)} \]

       5. +-commutative 66.4%

          \[\leadsto \frac{1}{a \cdot \left(-\left(1 + \left(b - \frac{\color{blue}{b + 2}}{a}\right)\right)\right)} \]

   13. Simplified 66.4%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + \left(b - \frac{b + 2}{a}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.4e+81)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (* b (- 1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.4e+81) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} 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 <= 6.4d+81) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    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 <= 6.4e+81) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (b * (1.0 - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.4e+81:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (b * (1.0 - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.4e+81)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	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 <= 6.4e+81)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (b * (1.0 - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.4e+81], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4e81

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 74.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 74.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 74.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 75.7%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 75.7%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 62.7%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 6.4e81 < 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%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 26.3%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 26.3%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 26.3%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 26.3%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 26.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 26.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 26.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 22.4%

      \[\leadsto \frac{1}{\color{blue}{\left(1 + \left(b + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)} + 1} \]
   9. Step-by-step derivation

      1. associate-+r+ 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + b\right) + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)} + 1} \]

      2. associate-*r* 22.4%

         \[\leadsto \frac{1}{\left(\left(1 + b\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(1 + b\right)}\right) + 1} \]

      3. distribute-rgt1-in 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot a + 1\right) \cdot \left(1 + b\right)} + 1} \]

      4. +-commutative 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 + -1 \cdot a\right)} \cdot \left(1 + b\right) + 1} \]

      5. neg-mul-1 22.4%

         \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-a\right)}\right) \cdot \left(1 + b\right) + 1} \]

      6. unsub-neg 22.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right)} \cdot \left(1 + b\right) + 1} \]

   10. Simplified 22.4%

       \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right) \cdot \left(1 + b\right)} + 1} \]

   11. Taylor expanded in b around inf 22.4%

       \[\leadsto \color{blue}{\frac{1}{b \cdot \left(1 - a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b - b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-15) (/ 1.0 (- 2.0 a)) (/ 1.0 (+ 2.0 (- b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (2.0 + (b - (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 <= 3d-15) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = 1.0d0 / (2.0d0 + (b - (b * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (2.0 + (b - (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-15:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 1.0 / (2.0 + (b - (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-15)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b - Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-15)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 1.0 / (2.0 + (b - (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-15], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b - N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3e-15

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.4%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.4%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 75.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 75.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 75.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 80.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 80.0%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 80.0%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 55.9%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
   9. Step-by-step derivation

      1. neg-mul-1 55.9%

         \[\leadsto \frac{1}{2 + \color{blue}{\left(-a\right)}} \]

      2. unsub-neg 55.9%

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   10. Simplified 55.9%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if 3e-15 < b

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.6%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.6%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.6%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 98.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 75.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 75.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 75.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.8%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 33.0%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 33.0%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 33.0%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 33.0%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 33.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 33.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 33.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 26.5%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + a \cdot \left(a \cdot \left(0.5 - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)}} \]

   9. Taylor expanded in b around inf 26.6%

      \[\leadsto \frac{1}{2 + \left(b + \color{blue}{a \cdot \left(b \cdot \left(0.5 \cdot a - 1\right)\right)}\right)} \]

   10. Taylor expanded in a around 0 20.3%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(-1 \cdot b\right)}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 20.3%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(-b\right)}\right)} \]

   12. Simplified 20.3%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \color{blue}{\left(-b\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b - b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.7% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9) (/ 1.0 (* a (- -1.0 b))) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9) {
		tmp = 1.0 / (a * (-1.0 - 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 <= (-1.9d0)) then
        tmp = 1.0d0 / (a * ((-1.0d0) - 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 <= -1.9) {
		tmp = 1.0 / (a * (-1.0 - b));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.9:
		tmp = 1.0 / (a * (-1.0 - b))
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9)
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 - 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 <= -1.9)
		tmp = 1.0 / (a * (-1.0 - b));
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9], N[(1.0 / N[(a * N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8999999999999999

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.4%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.4%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 98.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 1.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 1.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 1.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.4%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.4%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 98.4%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 98.4%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.8%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 61.9%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.9%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 100.0%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 19.4%

      \[\leadsto \frac{1}{\color{blue}{\left(1 + \left(b + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)} + 1} \]
   9. Step-by-step derivation

      1. associate-+r+ 19.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + b\right) + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)} + 1} \]

      2. associate-*r* 19.4%

         \[\leadsto \frac{1}{\left(\left(1 + b\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(1 + b\right)}\right) + 1} \]

      3. distribute-rgt1-in 19.4%

         \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot a + 1\right) \cdot \left(1 + b\right)} + 1} \]

      4. +-commutative 19.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 + -1 \cdot a\right)} \cdot \left(1 + b\right) + 1} \]

      5. neg-mul-1 19.4%

         \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-a\right)}\right) \cdot \left(1 + b\right) + 1} \]

      6. unsub-neg 19.4%

         \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right)} \cdot \left(1 + b\right) + 1} \]

   10. Simplified 19.4%

       \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right) \cdot \left(1 + b\right)} + 1} \]

   11. Taylor expanded in a around inf 19.4%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(a \cdot \left(1 + b\right)\right)}} \]
   12. Step-by-step derivation

       1. mul-1-neg 19.4%

          \[\leadsto \frac{1}{\color{blue}{-a \cdot \left(1 + b\right)}} \]

       2. distribute-rgt-neg-in 19.4%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-\left(1 + b\right)\right)}} \]

       3. mul-1-neg 19.4%

          \[\leadsto \frac{1}{a \cdot \color{blue}{\left(-1 \cdot \left(1 + b\right)\right)}} \]

       4. distribute-lft-in 19.4%

          \[\leadsto \frac{1}{a \cdot \color{blue}{\left(-1 \cdot 1 + -1 \cdot b\right)}} \]

       5. metadata-eval 19.4%

          \[\leadsto \frac{1}{a \cdot \left(\color{blue}{-1} + -1 \cdot b\right)} \]

       6. mul-1-neg 19.4%

          \[\leadsto \frac{1}{a \cdot \left(-1 + \color{blue}{\left(-b\right)}\right)} \]

       7. unsub-neg 19.4%

          \[\leadsto \frac{1}{a \cdot \color{blue}{\left(-1 - b\right)}} \]

   13. Simplified 19.4%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-1 - b\right)}} \]

   if -1.8999999999999999 < a

   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%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.5%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 99.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 99.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 99.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 54.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 54.9%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 54.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 54.1%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
   9. Step-by-step derivation

      1. *-commutative 54.1%

         \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

   10. Simplified 54.1%

       \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 - b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.8% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95) (/ 1.0 (- 2.0 a)) (/ 1.0 (* b (- 1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95) {
		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 <= 1.95d0) 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 <= 1.95) {
		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 <= 1.95:
		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 <= 1.95)
		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 <= 1.95)
		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, 1.95], 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 < 1.94999999999999996

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 75.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 75.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 75.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.5%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.5%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.5%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 79.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
   6. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   7. Simplified 79.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

   8. Taylor expanded in a around 0 55.1%

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
   9. Step-by-step derivation

      1. neg-mul-1 55.1%

         \[\leadsto \frac{1}{2 + \color{blue}{\left(-a\right)}} \]

      2. unsub-neg 55.1%

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   10. Simplified 55.1%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if 1.94999999999999996 < 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%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 76.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 76.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 76.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 28.5%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(e^{-a} + b \cdot e^{-a}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 28.5%

         \[\leadsto \frac{1}{\color{blue}{\left(e^{-a} + b \cdot e^{-a}\right) + 1}} \]

      2. distribute-rgt1-in 28.5%

         \[\leadsto \frac{1}{\color{blue}{\left(b + 1\right) \cdot e^{-a}} + 1} \]

      3. rec-exp 28.5%

         \[\leadsto \frac{1}{\left(b + 1\right) \cdot \color{blue}{\frac{1}{e^{a}}} + 1} \]

      4. associate-*r/ 28.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(b + 1\right) \cdot 1}{e^{a}}} + 1} \]

      5. *-rgt-identity 28.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{b + 1}}{e^{a}} + 1} \]

   7. Simplified 28.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{b + 1}{e^{a}} + 1}} \]

   8. Taylor expanded in a around 0 18.8%

      \[\leadsto \frac{1}{\color{blue}{\left(1 + \left(b + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)} + 1} \]
   9. Step-by-step derivation

      1. associate-+r+ 18.8%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(1 + b\right) + -1 \cdot \left(a \cdot \left(1 + b\right)\right)\right)} + 1} \]

      2. associate-*r* 18.8%

         \[\leadsto \frac{1}{\left(\left(1 + b\right) + \color{blue}{\left(-1 \cdot a\right) \cdot \left(1 + b\right)}\right) + 1} \]

      3. distribute-rgt1-in 18.8%

         \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot a + 1\right) \cdot \left(1 + b\right)} + 1} \]

      4. +-commutative 18.8%

         \[\leadsto \frac{1}{\color{blue}{\left(1 + -1 \cdot a\right)} \cdot \left(1 + b\right) + 1} \]

      5. neg-mul-1 18.8%

         \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-a\right)}\right) \cdot \left(1 + b\right) + 1} \]

      6. unsub-neg 18.8%

         \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right)} \cdot \left(1 + b\right) + 1} \]

   10. Simplified 18.8%

       \[\leadsto \frac{1}{\color{blue}{\left(1 - a\right) \cdot \left(1 + b\right)} + 1} \]

   11. Taylor expanded in b around inf 18.8%

       \[\leadsto \color{blue}{\frac{1}{b \cdot \left(1 - a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.3% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 75.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.6%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 99.6%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 99.6%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 99.9%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
6. Step-by-step derivation

   1. +-commutative 65.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

7. Simplified 65.9%

   \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

8. Taylor expanded in a around 0 41.3%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
9. Step-by-step derivation

   1. *-commutative 41.3%

      \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

10. Simplified 41.3%

    \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

11. Final simplification 41.3%

    \[\leadsto 0.5 + a \cdot 0.25 \]
12. Add Preprocessing

Alternative 20: 41.0% 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 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 75.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.6%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 99.6%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 99.6%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 99.9%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
6. Step-by-step derivation

   1. +-commutative 65.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

7. Simplified 65.9%

   \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

8. Taylor expanded in a around 0 41.7%

   \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
9. Step-by-step derivation

   1. neg-mul-1 41.7%

      \[\leadsto \frac{1}{2 + \color{blue}{\left(-a\right)}} \]

   2. unsub-neg 41.7%

      \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

10. Simplified 41.7%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

11. Final simplification 41.7%

    \[\leadsto \frac{1}{2 - a} \]
12. Add Preprocessing

Alternative 21: 40.1% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

double code(double a, double b) {
	return 0.5;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 75.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 75.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.6%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 99.6%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 99.6%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 99.9%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]
6. Step-by-step derivation

   1. +-commutative 65.9%

      \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

7. Simplified 65.9%

   \[\leadsto \frac{1}{\color{blue}{e^{-a} + 1}} \]

8. Taylor expanded in a around 0 40.5%

   \[\leadsto \color{blue}{0.5} \]

9. Final simplification 40.5%

   \[\leadsto 0.5 \]
10. Add Preprocessing

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 2024073 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))