Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 10.3s

Alternatives: 18

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 74.6%

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

   7. *-lft-identity 74.6%

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

   8. associate-*l/ 74.6%

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

   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: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. remove-double-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. div-sub 1.5%

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

      7. *-lft-identity 1.5%

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

      8. associate-*l/ 1.5%

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

      9. lft-mult-inverse 98.5%

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

      10. sub-neg 98.5%

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

      11. distribute-frac-neg 98.5%

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

      12. remove-double-neg 98.5%

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

      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 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 97.5%

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

      1. +-commutative 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 98.0%

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

Alternative 3: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-245}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-245) (/ (exp a) 2.0) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-245) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-245) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-245) {
		tmp = Math.exp(a) / 2.0;
	} 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) / 2.0
	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) / 2.0);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-245)
		tmp = exp(a) / 2.0;
	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] / 2.0), $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 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 99.5%

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

   4. Taylor expanded in a around 0 98.8%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]

   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 97.5%

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

      1. +-commutative 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-245}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + \left(\frac{b + 2}{a} - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.26e+168)
   (/ (exp a) 2.0)
   (/ 1.0 (* a (+ -1.0 (- (/ (+ b 2.0) a) b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.26e+168) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.26d+168) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (a * ((-1.0d0) + (((b + 2.0d0) / a) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.26e+168) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.26e+168:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.26e+168)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 + Float64(Float64(Float64(b + 2.0) / a) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.26e+168)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.26e+168], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 + N[(N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26000000000000006e168

   1. Initial program 99.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 72.6%

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

   4. Taylor expanded in a around 0 70.6%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]

   if 1.26000000000000006e168 < 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 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 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 26.2%

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

      1. +-commutative 26.2%

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

      2. distribute-rgt1-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 26.2%

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

      1. associate-+r+ 26.2%

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

      2. +-commutative 26.2%

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

      3. mul-1-neg 26.2%

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

      4. distribute-rgt-neg-in 26.2%

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

      5. mul-1-neg 26.2%

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

      6. distribute-lft-in 26.2%

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

      7. metadata-eval 26.2%

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

      8. mul-1-neg 26.2%

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

      9. unsub-neg 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in a around -inf 81.1%

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

       1. associate-*r* 81.1%

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

       2. neg-mul-1 81.1%

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

       3. mul-1-neg 81.1%

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

       4. unsub-neg 81.1%

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

       5. +-commutative 81.1%

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

   13. Simplified 81.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + \left(\frac{b + 2}{a} - b\right)\right)}\\ \end{array} \]
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 74.6%

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

   7. *-lft-identity 74.6%

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

   8. associate-*l/ 74.6%

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

   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: 65.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(-1 - b\right) + a \cdot \left(0.5 \cdot \left(1 + b\right) + -0.16666666666666666 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.2e-192)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (if (<= b 5e-239)
     (/ 1.0 (- (+ b 2.0) (* b (+ a (/ a b)))))
     (if (<= b 1.26e+168)
       (/
        1.0
        (+
         2.0
         (+
          b
          (*
           a
           (+
            (- -1.0 b)
            (*
             a
             (+
              (* 0.5 (+ 1.0 b))
              (* -0.16666666666666666 (* a (+ 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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 5e-239) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((0.5 * (1.0 + b)) + (-0.16666666666666666 * (a * (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 * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 5d-239) then
        tmp = 1.0d0 / ((b + 2.0d0) - (b * (a + (a / b))))
    else if (b <= 1.26d+168) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * (((-1.0d0) - b) + (a * ((0.5d0 * (1.0d0 + b)) + ((-0.16666666666666666d0) * (a * (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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 5e-239) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((0.5 * (1.0 + b)) + (-0.16666666666666666 * (a * (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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 5e-239:
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))))
	elif b <= 1.26e+168:
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((0.5 * (1.0 + b)) + (-0.16666666666666666 * (a * (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(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 5e-239)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * Float64(a + Float64(a / b)))));
	elseif (b <= 1.26e+168)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(-1.0 - b) + Float64(a * Float64(Float64(0.5 * Float64(1.0 + b)) + Float64(-0.16666666666666666 * Float64(a * 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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 5e-239)
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	elseif (b <= 1.26e+168)
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((0.5 * (1.0 + b)) + (-0.16666666666666666 * (a * (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[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-239], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e+168], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(-1.0 - b), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(1.0 + b), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(a * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -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/ 98.7%

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

      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 82.1%

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

      7. *-lft-identity 82.1%

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

      8. associate-*l/ 82.1%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      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 47.6%

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

   if -5.2000000000000003e-192 < b < 5e-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 59.2%

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

      7. *-lft-identity 59.2%

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

      8. associate-*l/ 59.2%

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

      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 77.6%

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

      1. +-commutative 77.6%

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

      2. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 59.8%

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

      1. associate-+r+ 59.8%

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

      2. +-commutative 59.8%

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

      3. mul-1-neg 59.8%

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

      4. distribute-rgt-neg-in 59.8%

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

      5. mul-1-neg 59.8%

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

      6. distribute-lft-in 59.8%

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

      7. metadata-eval 59.8%

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

      8. mul-1-neg 59.8%

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

      9. unsub-neg 59.8%

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

   10. Simplified 59.8%

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

   11. Taylor expanded in b around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. unsub-neg 96.4%

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

       3. neg-mul-1 96.4%

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

   13. Simplified 96.4%

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

   if 5e-239 < b < 1.26000000000000006e168

   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 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 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 75.1%

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

      1. +-commutative 75.1%

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

      2. distribute-rgt1-in 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in a around 0 66.3%

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

   if 1.26000000000000006e168 < 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 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 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 26.2%

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

      1. +-commutative 26.2%

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

      2. distribute-rgt1-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 26.2%

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

      1. associate-+r+ 26.2%

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

      2. +-commutative 26.2%

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

      3. mul-1-neg 26.2%

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

      4. distribute-rgt-neg-in 26.2%

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

      5. mul-1-neg 26.2%

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

      6. distribute-lft-in 26.2%

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

      7. metadata-eval 26.2%

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

      8. mul-1-neg 26.2%

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

      9. unsub-neg 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in a around -inf 81.1%

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

       1. associate-*r* 81.1%

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

       2. neg-mul-1 81.1%

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

       3. mul-1-neg 81.1%

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

       4. unsub-neg 81.1%

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

       5. +-commutative 81.1%

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

   13. Simplified 81.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(-1 - b\right) + a \cdot \left(0.5 \cdot \left(1 + b\right) + -0.16666666666666666 \cdot \left(a \cdot \left(1 + b\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(-1 - b\right) + a \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \left(a + \frac{a}{b}\right)\right) + 0.5 \cdot \left(1 + b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + \left(\frac{b + 2}{a} - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.15)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (if (<= b 1.26e+168)
     (/
      1.0
      (+
       2.0
       (+
        b
        (*
         a
         (+
          (- -1.0 b)
          (*
           a
           (+
            (* b (* -0.16666666666666666 (+ a (/ a b))))
            (* 0.5 (+ 1.0 b)))))))))
     (/ 1.0 (* a (+ -1.0 (- (/ (+ b 2.0) a) b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.15) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((b * (-0.16666666666666666 * (a + (a / b)))) + (0.5 * (1.0 + b))))))));
	} else {
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.15d0)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else if (b <= 1.26d+168) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * (((-1.0d0) - b) + (a * ((b * ((-0.16666666666666666d0) * (a + (a / b)))) + (0.5d0 * (1.0d0 + b))))))))
    else
        tmp = 1.0d0 / (a * ((-1.0d0) + (((b + 2.0d0) / a) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.15) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((b * (-0.16666666666666666 * (a + (a / b)))) + (0.5 * (1.0 + b))))))));
	} else {
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.15:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	elif b <= 1.26e+168:
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((b * (-0.16666666666666666 * (a + (a / b)))) + (0.5 * (1.0 + b))))))))
	else:
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.15)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	elseif (b <= 1.26e+168)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(-1.0 - b) + Float64(a * Float64(Float64(b * Float64(-0.16666666666666666 * Float64(a + Float64(a / b)))) + Float64(0.5 * Float64(1.0 + b)))))))));
	else
		tmp = Float64(1.0 / Float64(a * Float64(-1.0 + Float64(Float64(Float64(b + 2.0) / a) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.15)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	elseif (b <= 1.26e+168)
		tmp = 1.0 / (2.0 + (b + (a * ((-1.0 - b) + (a * ((b * (-0.16666666666666666 * (a + (a / b)))) + (0.5 * (1.0 + b))))))));
	else
		tmp = 1.0 / (a * (-1.0 + (((b + 2.0) / a) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.15], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e+168], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(-1.0 - b), $MachinePrecision] + N[(a * N[(N[(b * N[(-0.16666666666666666 * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(-1.0 + N[(N[(N[(b + 2.0), $MachinePrecision] / a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1499999999999999

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-*l/ 97.7%

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

      3. associate-/r/ 97.7%

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

      4. remove-double-neg 97.7%

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

      5. unsub-neg 97.7%

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

      6. div-sub 97.7%

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

      7. *-lft-identity 97.7%

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

      8. associate-*l/ 97.7%

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

      9. lft-mult-inverse 97.7%

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

      10. sub-neg 97.7%

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

      11. distribute-frac-neg 97.7%

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

      12. remove-double-neg 97.7%

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

      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 20.6%

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

      1. +-commutative 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in a around 0 20.6%

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

   if -1.1499999999999999 < b < 1.26000000000000006e168

   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 68.0%

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

      7. *-lft-identity 68.0%

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

      8. associate-*l/ 68.0%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 73.1%

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

      1. +-commutative 73.1%

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

      2. distribute-rgt1-in 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in a around 0 74.0%

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

   9. Taylor expanded in b around inf 78.2%

      \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(-1 \cdot \left(1 + b\right) + a \cdot \left(\color{blue}{b \cdot \left(-0.16666666666666666 \cdot a + -0.16666666666666666 \cdot \frac{a}{b}\right)} + 0.5 \cdot \left(1 + b\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-out 78.2%

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

   11. Simplified 78.2%

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

   if 1.26000000000000006e168 < 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 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 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 26.2%

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

      1. +-commutative 26.2%

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

      2. distribute-rgt1-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 26.2%

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

      1. associate-+r+ 26.2%

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

      2. +-commutative 26.2%

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

      3. mul-1-neg 26.2%

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

      4. distribute-rgt-neg-in 26.2%

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

      5. mul-1-neg 26.2%

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

      6. distribute-lft-in 26.2%

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

      7. metadata-eval 26.2%

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

      8. mul-1-neg 26.2%

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

      9. unsub-neg 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in a around -inf 81.1%

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

       1. associate-*r* 81.1%

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

       2. neg-mul-1 81.1%

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

       3. mul-1-neg 81.1%

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

       4. unsub-neg 81.1%

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

       5. +-commutative 81.1%

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

   13. Simplified 81.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(\left(-1 - b\right) + a \cdot \left(b \cdot \left(-0.16666666666666666 \cdot \left(a + \frac{a}{b}\right)\right) + 0.5 \cdot \left(1 + b\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + \left(\frac{b + 2}{a} - b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{2 + a \cdot t\_0}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(b \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666))))))
   (if (<= b -1e-191)
     (/ 1.0 (+ 2.0 (* a t_0)))
     (if (<= b 7.5e-79)
       (/ 1.0 (- (+ b 2.0) (* b (+ a (/ a b)))))
       (if (<= b 1.26e+168)
         (/ 1.0 (+ 2.0 (+ b (* a (* b t_0)))))
         (/ -1.0 (* a (+ 1.0 (- b (/ (+ b 2.0) a))))))))))

C

double code(double a, double b) {
	double t_0 = -1.0 + (a * (0.5 + (a * -0.16666666666666666)));
	double tmp;
	if (b <= -1e-191) {
		tmp = 1.0 / (2.0 + (a * t_0));
	} else if (b <= 7.5e-79) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * (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 = (-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0))))
    if (b <= (-1d-191)) then
        tmp = 1.0d0 / (2.0d0 + (a * t_0))
    else if (b <= 7.5d-79) then
        tmp = 1.0d0 / ((b + 2.0d0) - (b * (a + (a / b))))
    else if (b <= 1.26d+168) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * (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 = -1.0 + (a * (0.5 + (a * -0.16666666666666666)));
	double tmp;
	if (b <= -1e-191) {
		tmp = 1.0 / (2.0 + (a * t_0));
	} else if (b <= 7.5e-79) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (2.0 + (b + (a * (b * t_0))));
	} else {
		tmp = -1.0 / (a * (1.0 + (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -1.0 + (a * (0.5 + (a * -0.16666666666666666)))
	tmp = 0
	if b <= -1e-191:
		tmp = 1.0 / (2.0 + (a * t_0))
	elif b <= 7.5e-79:
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))))
	elif b <= 1.26e+168:
		tmp = 1.0 / (2.0 + (b + (a * (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(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))))
	tmp = 0.0
	if (b <= -1e-191)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * t_0)));
	elseif (b <= 7.5e-79)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * Float64(a + Float64(a / b)))));
	elseif (b <= 1.26e+168)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * 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 = -1.0 + (a * (0.5 + (a * -0.16666666666666666)));
	tmp = 0.0;
	if (b <= -1e-191)
		tmp = 1.0 / (2.0 + (a * t_0));
	elseif (b <= 7.5e-79)
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	elseif (b <= 1.26e+168)
		tmp = 1.0 / (2.0 + (b + (a * (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[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e-191], N[(1.0 / N[(2.0 + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-79], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e+168], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(b * t$95$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 < -1e-191

   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 82.1%

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

      7. *-lft-identity 82.1%

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

      8. associate-*l/ 82.1%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      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 47.6%

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

   if -1e-191 < b < 7.49999999999999969e-79

   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/ 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 66.3%

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

      7. *-lft-identity 66.3%

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

      8. associate-*l/ 66.3%

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

      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 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 87.6%

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

      1. +-commutative 87.6%

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

      2. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 65.8%

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

      1. associate-+r+ 65.8%

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

      2. +-commutative 65.8%

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

      3. mul-1-neg 65.8%

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

      4. distribute-rgt-neg-in 65.8%

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

      5. mul-1-neg 65.8%

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

      6. distribute-lft-in 65.8%

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

      7. metadata-eval 65.8%

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

      8. mul-1-neg 65.8%

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

      9. unsub-neg 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in b around inf 93.2%

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

       1. mul-1-neg 93.2%

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

       2. unsub-neg 93.2%

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

       3. neg-mul-1 93.2%

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

   13. Simplified 93.2%

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

   if 7.49999999999999969e-79 < b < 1.26000000000000006e168

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-*l/ 98.1%

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

      3. associate-/r/ 98.1%

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

      4. remove-double-neg 98.1%

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

      5. unsub-neg 98.1%

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

      6. div-sub 74.1%

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

      7. *-lft-identity 74.1%

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

      8. associate-*l/ 74.1%

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

      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 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 56.7%

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

      1. +-commutative 56.7%

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

      2. distribute-rgt1-in 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in a around 0 47.9%

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

   9. Taylor expanded in b around inf 47.8%

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

   if 1.26000000000000006e168 < 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 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 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 26.2%

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

      1. +-commutative 26.2%

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

      2. distribute-rgt1-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 26.2%

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

      1. associate-+r+ 26.2%

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

      2. +-commutative 26.2%

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

      3. mul-1-neg 26.2%

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

      4. distribute-rgt-neg-in 26.2%

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

      5. mul-1-neg 26.2%

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

      6. distribute-lft-in 26.2%

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

      7. metadata-eval 26.2%

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

      8. mul-1-neg 26.2%

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

      9. unsub-neg 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in a around -inf 81.1%

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

       1. associate-*r* 81.1%

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

       2. neg-mul-1 81.1%

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

       3. mul-1-neg 81.1%

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

       4. unsub-neg 81.1%

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

       5. +-commutative 81.1%

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

   13. Simplified 81.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(b \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + t\_0}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (if (<= b -5.2e-192)
     (/ 1.0 (+ 2.0 t_0))
     (if (<= b 6.4e-9)
       (/ 1.0 (- (+ b 2.0) (* b (+ a (/ a b)))))
       (if (<= b 1.26e+168)
         (/ 1.0 (* b (+ 1.0 t_0)))
         (/ -1.0 (* a (+ 1.0 (- b (/ (+ b 2.0) a))))))))))

C

double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	double tmp;
	if (b <= -5.2e-192) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 6.4e-9) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (b * (1.0 + 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 * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))
    if (b <= (-5.2d-192)) then
        tmp = 1.0d0 / (2.0d0 + t_0)
    else if (b <= 6.4d-9) then
        tmp = 1.0d0 / ((b + 2.0d0) - (b * (a + (a / b))))
    else if (b <= 1.26d+168) then
        tmp = 1.0d0 / (b * (1.0d0 + 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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	double tmp;
	if (b <= -5.2e-192) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 6.4e-9) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 1.26e+168) {
		tmp = 1.0 / (b * (1.0 + t_0));
	} else {
		tmp = -1.0 / (a * (1.0 + (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))
	tmp = 0
	if b <= -5.2e-192:
		tmp = 1.0 / (2.0 + t_0)
	elif b <= 6.4e-9:
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))))
	elif b <= 1.26e+168:
		tmp = 1.0 / (b * (1.0 + 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(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))
	tmp = 0.0
	if (b <= -5.2e-192)
		tmp = Float64(1.0 / Float64(2.0 + t_0));
	elseif (b <= 6.4e-9)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * Float64(a + Float64(a / b)))));
	elseif (b <= 1.26e+168)
		tmp = Float64(1.0 / Float64(b * Float64(1.0 + 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 * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	tmp = 0.0;
	if (b <= -5.2e-192)
		tmp = 1.0 / (2.0 + t_0);
	elseif (b <= 6.4e-9)
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	elseif (b <= 1.26e+168)
		tmp = 1.0 / (b * (1.0 + 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[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.2e-192], N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.4e-9], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e+168], N[(1.0 / N[(b * N[(1.0 + 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 < -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/ 98.7%

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

      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 82.1%

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

      7. *-lft-identity 82.1%

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

      8. associate-*l/ 82.1%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      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 47.6%

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

   if -5.2000000000000003e-192 < b < 6.40000000000000023e-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 69.1%

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

      7. *-lft-identity 69.1%

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

      8. associate-*l/ 69.1%

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

      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 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%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 68.1%

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

      1. associate-+r+ 68.1%

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

      2. +-commutative 68.1%

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

      3. mul-1-neg 68.1%

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

      4. distribute-rgt-neg-in 68.1%

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

      5. mul-1-neg 68.1%

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

      6. distribute-lft-in 68.1%

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

      7. metadata-eval 68.1%

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

      8. mul-1-neg 68.1%

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

      9. unsub-neg 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in b around inf 91.6%

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

       1. mul-1-neg 91.6%

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

       2. unsub-neg 91.6%

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

       3. neg-mul-1 91.6%

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

   13. Simplified 91.6%

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

   if 6.40000000000000023e-9 < b < 1.26000000000000006e168

   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 69.4%

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

      7. *-lft-identity 69.4%

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

      8. associate-*l/ 69.4%

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

      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 35.0%

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

      1. +-commutative 35.0%

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

      2. distribute-rgt1-in 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in a around 0 29.9%

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

   9. Taylor expanded in b around inf 28.2%

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

   if 1.26000000000000006e168 < 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 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 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 26.2%

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

      1. +-commutative 26.2%

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

      2. distribute-rgt1-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 26.2%

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

      1. associate-+r+ 26.2%

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

      2. +-commutative 26.2%

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

      3. mul-1-neg 26.2%

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

      4. distribute-rgt-neg-in 26.2%

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

      5. mul-1-neg 26.2%

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

      6. distribute-lft-in 26.2%

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

      7. metadata-eval 26.2%

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

      8. mul-1-neg 26.2%

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

      9. unsub-neg 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in a around -inf 81.1%

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

       1. associate-*r* 81.1%

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

       2. neg-mul-1 81.1%

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

       3. mul-1-neg 81.1%

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

       4. unsub-neg 81.1%

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

       5. +-commutative 81.1%

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

   13. Simplified 81.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))))
   (if (<= b -5.2e-192)
     t_0
     (if (<= b 1.7e-237)
       (/ 1.0 (- (+ b 2.0) (* b (+ a (/ a b)))))
       (if (<= b 7.6e+76) t_0 (/ -1.0 (* a (+ 1.0 (- b (/ (+ b 2.0) a))))))))))

C

double code(double a, double b) {
	double t_0 = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	double tmp;
	if (b <= -5.2e-192) {
		tmp = t_0;
	} else if (b <= 1.7e-237) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 7.6e+76) {
		tmp = 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 = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    if (b <= (-5.2d-192)) then
        tmp = t_0
    else if (b <= 1.7d-237) then
        tmp = 1.0d0 / ((b + 2.0d0) - (b * (a + (a / b))))
    else if (b <= 7.6d+76) then
        tmp = 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 = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	double tmp;
	if (b <= -5.2e-192) {
		tmp = t_0;
	} else if (b <= 1.7e-237) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	} else if (b <= 7.6e+76) {
		tmp = t_0;
	} else {
		tmp = -1.0 / (a * (1.0 + (b - ((b + 2.0) / a))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	tmp = 0
	if b <= -5.2e-192:
		tmp = t_0
	elif b <= 1.7e-237:
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))))
	elif b <= 7.6e+76:
		tmp = t_0
	else:
		tmp = -1.0 / (a * (1.0 + (b - ((b + 2.0) / a))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))))
	tmp = 0.0
	if (b <= -5.2e-192)
		tmp = t_0;
	elseif (b <= 1.7e-237)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * Float64(a + Float64(a / b)))));
	elseif (b <= 7.6e+76)
		tmp = 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 = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	tmp = 0.0;
	if (b <= -5.2e-192)
		tmp = t_0;
	elseif (b <= 1.7e-237)
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / b))));
	elseif (b <= 7.6e+76)
		tmp = 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[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.2e-192], t$95$0, If[LessEqual[b, 1.7e-237], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e+76], t$95$0, 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 or 1.7000000000000001e-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 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 99.4%

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

      10. sub-neg 99.4%

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

      11. distribute-frac-neg 99.4%

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

      12. remove-double-neg 99.4%

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

      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 68.8%

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

      1. +-commutative 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in a around 0 60.2%

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

   if -5.2000000000000003e-192 < b < 1.7000000000000001e-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 59.2%

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

      7. *-lft-identity 59.2%

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

      8. associate-*l/ 59.2%

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

      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 77.6%

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

      1. +-commutative 77.6%

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

      2. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 59.8%

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

      1. associate-+r+ 59.8%

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

      2. +-commutative 59.8%

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

      3. mul-1-neg 59.8%

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

      4. distribute-rgt-neg-in 59.8%

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

      5. mul-1-neg 59.8%

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

      6. distribute-lft-in 59.8%

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

      7. metadata-eval 59.8%

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

      8. mul-1-neg 59.8%

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

      9. unsub-neg 59.8%

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

   10. Simplified 59.8%

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

   11. Taylor expanded in b around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. unsub-neg 96.4%

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

       3. neg-mul-1 96.4%

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

   13. Simplified 96.4%

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

   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 78.0%

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

      7. *-lft-identity 78.0%

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

      8. associate-*l/ 78.0%

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

      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 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 27.3%

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

      1. +-commutative 27.3%

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

      2. distribute-rgt1-in 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in a around 0 21.7%

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

      1. associate-+r+ 21.7%

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

      2. +-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

      5. mul-1-neg 21.7%

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

      6. distribute-lft-in 21.7%

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

      7. metadata-eval 21.7%

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

      8. mul-1-neg 21.7%

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

      9. unsub-neg 21.7%

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

   10. Simplified 21.7%

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

   11. Taylor expanded in a around -inf 62.0%

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

       1. associate-*r* 62.0%

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

       2. neg-mul-1 62.0%

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

       3. mul-1-neg 62.0%

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

       4. unsub-neg 62.0%

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

       5. +-commutative 62.0%

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

   13. Simplified 62.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.8% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 10^{-13}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.2e-192)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (if (<= b 1e-13)
     (/ 1.0 (- (+ b 2.0) (* b (+ a (/ a 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) + -1.0)));
	} else if (b <= 1e-13) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / 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) + (-1.0d0))))
    else if (b <= 1d-13) then
        tmp = 1.0d0 / ((b + 2.0d0) - (b * (a + (a / 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) + -1.0)));
	} else if (b <= 1e-13) {
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / 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) + -1.0)))
	elif b <= 1e-13:
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / 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 * 0.5) + -1.0))));
	elseif (b <= 1e-13)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * Float64(a + Float64(a / 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) + -1.0)));
	elseif (b <= 1e-13)
		tmp = 1.0 / ((b + 2.0) - (b * (a + (a / 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 * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-13], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * N[(a + N[(a / b), $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/ 98.7%

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

      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 82.1%

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

      7. *-lft-identity 82.1%

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

      8. associate-*l/ 82.1%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      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 42.7%

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

   if -5.2000000000000003e-192 < b < 1e-13

   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 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 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 89.4%

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

      1. +-commutative 89.4%

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

      2. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 69.1%

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

      1. associate-+r+ 69.1%

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

      2. +-commutative 69.1%

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

      3. mul-1-neg 69.1%

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

      4. distribute-rgt-neg-in 69.1%

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

      5. mul-1-neg 69.1%

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

      6. distribute-lft-in 69.1%

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

      7. metadata-eval 69.1%

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

      8. mul-1-neg 69.1%

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

      9. unsub-neg 69.1%

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

   10. Simplified 69.1%

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

   11. Taylor expanded in b around inf 93.2%

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

       1. mul-1-neg 93.2%

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

       2. unsub-neg 93.2%

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

       3. neg-mul-1 93.2%

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

   13. Simplified 93.2%

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

   if 1e-13 < 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 74.3%

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

      7. *-lft-identity 74.3%

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

      8. associate-*l/ 74.3%

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

      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 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 33.5%

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

      1. +-commutative 33.5%

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

      2. distribute-rgt1-in 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in a around 0 19.6%

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

      1. associate-+r+ 19.6%

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

      2. +-commutative 19.6%

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

      3. mul-1-neg 19.6%

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

      4. distribute-rgt-neg-in 19.6%

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

      5. mul-1-neg 19.6%

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

      6. distribute-lft-in 19.6%

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

      7. metadata-eval 19.6%

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

      8. mul-1-neg 19.6%

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

      9. unsub-neg 19.6%

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

   10. Simplified 19.6%

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

   11. Taylor expanded in a around -inf 46.8%

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

       1. associate-*r* 46.8%

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

       2. neg-mul-1 46.8%

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

       3. mul-1-neg 46.8%

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

       4. unsub-neg 46.8%

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

       5. +-commutative 46.8%

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

   13. Simplified 46.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{elif}\;b \leq 10^{-13}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + \left(b - \frac{b + 2}{a}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.7% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.6e+76)
		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(Float64(Float64(b + 2.0) / a) - b))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000049e76

   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 73.8%

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

      7. *-lft-identity 73.8%

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

      8. associate-*l/ 73.8%

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

      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 76.3%

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

      1. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in a around 0 62.6%

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

   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 78.0%

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

      7. *-lft-identity 78.0%

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

      8. associate-*l/ 78.0%

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

      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 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 27.3%

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

      1. +-commutative 27.3%

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

      2. distribute-rgt1-in 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in a around 0 21.7%

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

      1. associate-+r+ 21.7%

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

      2. +-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

      5. mul-1-neg 21.7%

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

      6. distribute-lft-in 21.7%

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

      7. metadata-eval 21.7%

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

      8. mul-1-neg 21.7%

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

      9. unsub-neg 21.7%

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

   10. Simplified 21.7%

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

   11. Taylor expanded in a around -inf 62.0%

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

       1. associate-*r* 62.0%

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

       2. neg-mul-1 62.0%

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

       3. mul-1-neg 62.0%

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

       4. unsub-neg 62.0%

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

       5. +-commutative 62.0%

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

   13. Simplified 62.0%

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

4. Final simplification 62.5%

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

Alternative 13: 55.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+78)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (- (+ b 2.0) (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+78) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / ((b + 2.0) - (b * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.5d+78) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / ((b + 2.0d0) - (b * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+78) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / ((b + 2.0) - (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.5e+78:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / ((b + 2.0) - (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+78)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.5e+78)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / ((b + 2.0) - (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+78], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e78

   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 73.8%

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

      7. *-lft-identity 73.8%

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

      8. associate-*l/ 73.8%

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

      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 76.3%

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

      1. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in a around 0 62.6%

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

   if 3.5000000000000001e78 < 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 78.0%

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

      7. *-lft-identity 78.0%

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

      8. associate-*l/ 78.0%

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

      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 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 27.3%

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

      1. +-commutative 27.3%

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

      2. distribute-rgt1-in 27.3%

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

   7. Simplified 27.3%

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

   8. Taylor expanded in a around 0 21.7%

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

      1. associate-+r+ 21.7%

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

      2. +-commutative 21.7%

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

      3. mul-1-neg 21.7%

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

      4. distribute-rgt-neg-in 21.7%

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

      5. mul-1-neg 21.7%

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

      6. distribute-lft-in 21.7%

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

      7. metadata-eval 21.7%

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

      8. mul-1-neg 21.7%

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

      9. unsub-neg 21.7%

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

   10. Simplified 21.7%

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

   11. Taylor expanded in b around inf 21.7%

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

       1. associate-*r* 21.7%

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

       2. neg-mul-1 21.7%

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

   13. Simplified 21.7%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.0% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.7e-15) (/ 1.0 (- 2.0 a)) (/ 1.0 (- (+ b 2.0) (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.7e-15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / ((b + 2.0) - (b * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.7d-15) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = 1.0d0 / ((b + 2.0d0) - (b * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.7e-15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / ((b + 2.0) - (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.7e-15:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 1.0 / ((b + 2.0) - (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.7e-15)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) - Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.7e-15)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 1.0 / ((b + 2.0) - (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.7e-15], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] - N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000009e-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 74.7%

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

      7. *-lft-identity 74.7%

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

      8. associate-*l/ 74.7%

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

      9. lft-mult-inverse 99.4%

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

      10. sub-neg 99.4%

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

      11. distribute-frac-neg 99.4%

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

      12. remove-double-neg 99.4%

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

      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 80.7%

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

      1. +-commutative 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in a around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. unsub-neg 55.6%

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

   10. Simplified 55.6%

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

   if 2.70000000000000009e-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 74.3%

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

      7. *-lft-identity 74.3%

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

      8. associate-*l/ 74.3%

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

      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 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 33.5%

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

      1. +-commutative 33.5%

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

      2. distribute-rgt1-in 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in a around 0 19.6%

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

      1. associate-+r+ 19.6%

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

      2. +-commutative 19.6%

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

      3. mul-1-neg 19.6%

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

      4. distribute-rgt-neg-in 19.6%

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

      5. mul-1-neg 19.6%

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

      6. distribute-lft-in 19.6%

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

      7. metadata-eval 19.6%

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

      8. mul-1-neg 19.6%

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

      9. unsub-neg 19.6%

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

   10. Simplified 19.6%

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

   11. Taylor expanded in b around inf 19.8%

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

       1. associate-*r* 19.8%

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

       2. neg-mul-1 19.8%

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

   13. Simplified 19.8%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) - b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.6% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e-5) (/ -1.0 (* a (+ 1.0 b))) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-5) {
		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 <= (-4.2d-5)) 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 <= -4.2e-5) {
		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 <= -4.2e-5:
		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 <= -4.2e-5)
		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 <= -4.2e-5)
		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, -4.2e-5], 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 < -4.19999999999999977e-5

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. remove-double-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. div-sub 3.0%

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

      7. *-lft-identity 3.0%

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

      8. associate-*l/ 3.0%

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

      9. lft-mult-inverse 98.5%

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

      10. sub-neg 98.5%

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

      11. distribute-frac-neg 98.5%

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

      12. remove-double-neg 98.5%

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

      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 60.7%

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

      1. +-commutative 60.7%

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

      2. distribute-rgt1-in 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in a around 0 18.7%

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

      1. associate-+r+ 18.7%

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

      2. +-commutative 18.7%

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

      3. mul-1-neg 18.7%

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

      4. distribute-rgt-neg-in 18.7%

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

      5. mul-1-neg 18.7%

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

      6. distribute-lft-in 18.7%

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

      7. metadata-eval 18.7%

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

      8. mul-1-neg 18.7%

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

      9. unsub-neg 18.7%

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

   10. Simplified 18.7%

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

   11. Taylor expanded in a around inf 18.7%

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

   if -4.19999999999999977e-5 < 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/ 99.4%

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

      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 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 55.1%

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

      1. +-commutative 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in a around 0 54.2%

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

      1. *-commutative 54.2%

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

   10. Simplified 54.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 74.6%

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

   7. *-lft-identity 74.6%

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

   8. associate-*l/ 74.6%

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

   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 66.1%

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

   1. +-commutative 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in a around 0 40.8%

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

   1. *-commutative 40.8%

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

10. Simplified 40.8%

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

11. Final simplification 40.8%

    \[\leadsto 0.5 + a \cdot 0.25 \]
12. Add Preprocessing

Alternative 17: 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 74.6%

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

   7. *-lft-identity 74.6%

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

   8. associate-*l/ 74.6%

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

   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 66.1%

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

   1. +-commutative 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in a around 0 41.1%

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

   1. neg-mul-1 41.1%

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

   2. unsub-neg 41.1%

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

10. Simplified 41.1%

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

11. Final simplification 41.1%

    \[\leadsto \frac{1}{2 - a} \]
12. Add Preprocessing

Alternative 18: 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 74.6%

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

   7. *-lft-identity 74.6%

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

   8. associate-*l/ 74.6%

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

   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 66.1%

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

   1. +-commutative 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in a around 0 39.8%

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

9. Final simplification 39.8%

   \[\leadsto 0.5 \]
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))))