Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 8.2s

Alternatives: 16

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   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. Add Preprocessing

Alternative 2: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -30500:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -30500.0) (/ (exp a) b) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -30500.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-30500.0d0)) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -30500.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -30500.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -30500.0)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -30500.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -30500.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -30500

   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 0.0%

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

      7. *-lft-identity 0.0%

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

      8. associate-*l/ 0.0%

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

      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 68.1%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 100.0%

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

   if -30500 < a

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-*l/ 97.8%

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

      3. associate-/r/ 97.8%

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

      4. remove-double-neg 97.8%

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

      5. unsub-neg 97.8%

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

      6. div-sub 97.3%

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

      7. *-lft-identity 97.3%

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

      8. associate-*l/ 97.3%

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

      9. lft-mult-inverse 98.9%

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

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

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

      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 97.7%

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

Alternative 3: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -550:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -550.0)
   (/ (exp a) b)
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -550.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-550.0d0)) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -550.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -550.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -550.0)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -550.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -550.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -550

   1. Initial program 97.3%

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

      1. *-lft-identity 97.3%

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

      2. associate-*l/ 97.3%

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

      3. associate-/r/ 97.3%

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

      4. remove-double-neg 97.3%

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

      5. unsub-neg 97.3%

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

      6. div-sub 0.0%

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

      7. *-lft-identity 0.0%

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

      8. associate-*l/ 0.0%

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

      9. lft-mult-inverse 97.3%

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

      10. sub-neg 97.3%

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

      11. distribute-frac-neg 97.3%

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

      12. remove-double-neg 97.3%

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

      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 65.3%

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

      1. distribute-rgt1-in 97.4%

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

      2. rec-exp 97.4%

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

      3. associate-*r/ 97.4%

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

      4. *-rgt-identity 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in b around inf 97.4%

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

   if -550 < a

   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 98.9%

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

      7. *-lft-identity 98.9%

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

      8. associate-*l/ 98.9%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 98.2%

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

   6. Taylor expanded in b around 0 65.5%

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

      1. *-commutative 65.5%

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

   8. Simplified 65.5%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -400:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - b \cdot \left(0.16666666666666666 \cdot \left(a + \frac{a}{b}\right)\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -400.0)
   0.5
   (if (<= b 1.05e+103)
     (/
      1.0
      (+
       2.0
       (+
        b
        (*
         a
         (+
          (*
           a
           (+
            (- 0.5 (* b (* 0.16666666666666666 (+ a (/ a b)))))
            (- b (* b 0.5))))
          (- -1.0 b))))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -400.0) {
		tmp = 0.5;
	} else if (b <= 1.05e+103) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a + (a / b))))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-400.0d0)) then
        tmp = 0.5d0
    else if (b <= 1.05d+103) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * ((0.5d0 - (b * (0.16666666666666666d0 * (a + (a / b))))) + (b - (b * 0.5d0)))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -400.0) {
		tmp = 0.5;
	} else if (b <= 1.05e+103) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a + (a / b))))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -400.0:
		tmp = 0.5
	elif b <= 1.05e+103:
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a + (a / b))))) + (b - (b * 0.5)))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -400.0)
		tmp = 0.5;
	elseif (b <= 1.05e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(Float64(0.5 - Float64(b * Float64(0.16666666666666666 * Float64(a + Float64(a / b))))) + Float64(b - Float64(b * 0.5)))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -400.0)
		tmp = 0.5;
	elseif (b <= 1.05e+103)
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a + (a / b))))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -400.0], 0.5, If[LessEqual[b, 1.05e+103], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(N[(0.5 - N[(b * N[(0.16666666666666666 * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b - N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -400

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-*l/ 92.7%

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

      3. associate-/r/ 92.7%

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

      4. remove-double-neg 92.7%

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

      5. unsub-neg 92.7%

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

      6. div-sub 92.7%

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

      7. *-lft-identity 92.7%

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

      8. associate-*l/ 92.7%

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

      9. lft-mult-inverse 96.4%

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

      10. sub-neg 96.4%

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

      11. distribute-frac-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -400 < b < 1.0500000000000001e103

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      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 64.7%

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

      7. *-lft-identity 64.7%

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

      8. associate-*l/ 64.7%

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

      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 79.4%

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

      1. distribute-rgt1-in 94.8%

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

      2. rec-exp 94.8%

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

      3. associate-*r/ 94.8%

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

      4. *-rgt-identity 94.8%

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

      5. +-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around 0 83.5%

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

   9. Taylor expanded in b around inf 84.7%

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

       1. distribute-lft-out 84.7%

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

   11. Simplified 84.7%

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

   if 1.0500000000000001e103 < 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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -400:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - b \cdot \left(0.16666666666666666 \cdot \left(a + \frac{a}{b}\right)\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - b \cdot \left(0.16666666666666666 \cdot \frac{a}{b}\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4100.0)
   0.5
   (if (<= b 5.2e+102)
     (/
      1.0
      (+
       2.0
       (+
        b
        (*
         a
         (+
          (*
           a
           (+ (- 0.5 (* b (* 0.16666666666666666 (/ a b)))) (- b (* b 0.5))))
          (- -1.0 b))))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4100.0) {
		tmp = 0.5;
	} else if (b <= 5.2e+102) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a / b)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4100.0d0)) then
        tmp = 0.5d0
    else if (b <= 5.2d+102) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * ((0.5d0 - (b * (0.16666666666666666d0 * (a / b)))) + (b - (b * 0.5d0)))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4100.0) {
		tmp = 0.5;
	} else if (b <= 5.2e+102) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a / b)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4100.0:
		tmp = 0.5
	elif b <= 5.2e+102:
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a / b)))) + (b - (b * 0.5)))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4100.0)
		tmp = 0.5;
	elseif (b <= 5.2e+102)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(Float64(0.5 - Float64(b * Float64(0.16666666666666666 * Float64(a / b)))) + Float64(b - Float64(b * 0.5)))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4100.0)
		tmp = 0.5;
	elseif (b <= 5.2e+102)
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (b * (0.16666666666666666 * (a / b)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4100.0], 0.5, If[LessEqual[b, 5.2e+102], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(N[(0.5 - N[(b * N[(0.16666666666666666 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b - N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4100

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-*l/ 92.7%

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

      3. associate-/r/ 92.7%

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

      4. remove-double-neg 92.7%

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

      5. unsub-neg 92.7%

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

      6. div-sub 92.7%

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

      7. *-lft-identity 92.7%

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

      8. associate-*l/ 92.7%

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

      9. lft-mult-inverse 96.4%

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

      10. sub-neg 96.4%

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

      11. distribute-frac-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -4100 < b < 5.20000000000000013e102

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      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 64.7%

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

      7. *-lft-identity 64.7%

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

      8. associate-*l/ 64.7%

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

      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 79.4%

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

      1. distribute-rgt1-in 94.8%

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

      2. rec-exp 94.8%

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

      3. associate-*r/ 94.8%

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

      4. *-rgt-identity 94.8%

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

      5. +-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around 0 83.5%

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

   9. Taylor expanded in b around inf 84.7%

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

       1. distribute-lft-out 84.7%

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

   11. Simplified 84.7%

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

   12. Taylor expanded in b around 0 84.7%

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

   if 5.20000000000000013e102 < 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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - b \cdot \left(0.16666666666666666 \cdot \frac{a}{b}\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -60:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - a \cdot \left(0.16666666666666666 + b \cdot -0.8333333333333334\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -60.0)
   0.5
   (if (<= b 1e+103)
     (/
      1.0
      (+
       2.0
       (+
        b
        (*
         a
         (+
          (*
           a
           (+
            (- 0.5 (* a (+ 0.16666666666666666 (* b -0.8333333333333334))))
            (- b (* b 0.5))))
          (- -1.0 b))))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -60.0) {
		tmp = 0.5;
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * (0.16666666666666666 + (b * -0.8333333333333334)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-60.0d0)) then
        tmp = 0.5d0
    else if (b <= 1d+103) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * ((0.5d0 - (a * (0.16666666666666666d0 + (b * (-0.8333333333333334d0))))) + (b - (b * 0.5d0)))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -60.0) {
		tmp = 0.5;
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * (0.16666666666666666 + (b * -0.8333333333333334)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -60.0:
		tmp = 0.5
	elif b <= 1e+103:
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * (0.16666666666666666 + (b * -0.8333333333333334)))) + (b - (b * 0.5)))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -60.0)
		tmp = 0.5;
	elseif (b <= 1e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(Float64(0.5 - Float64(a * Float64(0.16666666666666666 + Float64(b * -0.8333333333333334)))) + Float64(b - Float64(b * 0.5)))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -60.0)
		tmp = 0.5;
	elseif (b <= 1e+103)
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * (0.16666666666666666 + (b * -0.8333333333333334)))) + (b - (b * 0.5)))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -60.0], 0.5, If[LessEqual[b, 1e+103], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(N[(0.5 - N[(a * N[(0.16666666666666666 + N[(b * -0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b - N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -60

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-*l/ 92.7%

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

      3. associate-/r/ 92.7%

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

      4. remove-double-neg 92.7%

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

      5. unsub-neg 92.7%

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

      6. div-sub 92.7%

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

      7. *-lft-identity 92.7%

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

      8. associate-*l/ 92.7%

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

      9. lft-mult-inverse 96.4%

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

      10. sub-neg 96.4%

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

      11. distribute-frac-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -60 < b < 1e103

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      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 64.7%

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

      7. *-lft-identity 64.7%

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

      8. associate-*l/ 64.7%

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

      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 79.4%

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

      1. distribute-rgt1-in 94.8%

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

      2. rec-exp 94.8%

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

      3. associate-*r/ 94.8%

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

      4. *-rgt-identity 94.8%

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

      5. +-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around 0 83.5%

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

      1. *-un-lft-identity 83.5%

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

      2. +-commutative 83.5%

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

      3. distribute-rgt-out 83.5%

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

      4. fma-define 83.5%

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

      5. metadata-eval 83.5%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, \color{blue}{-0.3333333333333333}, -1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      6. add-sqr-sqrt 45.9%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \color{blue}{\sqrt{-1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)} \cdot \sqrt{-1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)}}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      7. sqrt-unprod 83.5%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \color{blue}{\sqrt{\left(-1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)\right) \cdot \left(-1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)\right)}}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      8. mul-1-neg 83.5%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \sqrt{\color{blue}{\left(-\left(-1 \cdot b + 0.5 \cdot b\right)\right)} \cdot \left(-1 \cdot \left(-1 \cdot b + 0.5 \cdot b\right)\right)}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      9. mul-1-neg 83.5%

         \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \sqrt{\left(-\left(-1 \cdot b + 0.5 \cdot b\right)\right) \cdot \color{blue}{\left(-\left(-1 \cdot b + 0.5 \cdot b\right)\right)}}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      10. sqr-neg 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \sqrt{\color{blue}{\left(-1 \cdot b + 0.5 \cdot b\right) \cdot \left(-1 \cdot b + 0.5 \cdot b\right)}}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      11. sqrt-unprod 37.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \color{blue}{\sqrt{-1 \cdot b + 0.5 \cdot b} \cdot \sqrt{-1 \cdot b + 0.5 \cdot b}}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      12. add-sqr-sqrt 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \color{blue}{-1 \cdot b + 0.5 \cdot b}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      13. distribute-rgt-out 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, \color{blue}{b \cdot \left(-1 + 0.5\right)}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

      14. metadata-eval 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + 1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, b \cdot \color{blue}{-0.5}\right)\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

   10. Applied egg-rr 83.5%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{1 \cdot \mathsf{fma}\left(b, -0.3333333333333333, b \cdot -0.5\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\mathsf{fma}\left(b, -0.3333333333333333, b \cdot -0.5\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       2. fma-undefine 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\left(b \cdot -0.3333333333333333 + b \cdot -0.5\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       3. distribute-lft-out 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{b \cdot \left(-0.3333333333333333 + -0.5\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       4. metadata-eval 83.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + b \cdot \color{blue}{-0.8333333333333334}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 83.5%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{b \cdot -0.8333333333333334}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

   if 1e103 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -60:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - a \cdot \left(0.16666666666666666 + b \cdot -0.8333333333333334\right)\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -35:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - a \cdot 0.16666666666666666\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -35.0)
   0.5
   (if (<= b 3.3e+102)
     (/
      1.0
      (+
       2.0
       (+
        b
        (*
         a
         (+
          (* a (+ (- 0.5 (* a 0.16666666666666666)) (- b (* b 0.5))))
          (- -1.0 b))))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -35.0) {
		tmp = 0.5;
	} else if (b <= 3.3e+102) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * 0.16666666666666666)) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-35.0d0)) then
        tmp = 0.5d0
    else if (b <= 3.3d+102) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((a * ((0.5d0 - (a * 0.16666666666666666d0)) + (b - (b * 0.5d0)))) + ((-1.0d0) - b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -35.0) {
		tmp = 0.5;
	} else if (b <= 3.3e+102) {
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * 0.16666666666666666)) + (b - (b * 0.5)))) + (-1.0 - b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -35.0:
		tmp = 0.5
	elif b <= 3.3e+102:
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * 0.16666666666666666)) + (b - (b * 0.5)))) + (-1.0 - b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -35.0)
		tmp = 0.5;
	elseif (b <= 3.3e+102)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(Float64(a * Float64(Float64(0.5 - Float64(a * 0.16666666666666666)) + Float64(b - Float64(b * 0.5)))) + Float64(-1.0 - b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -35.0)
		tmp = 0.5;
	elseif (b <= 3.3e+102)
		tmp = 1.0 / (2.0 + (b + (a * ((a * ((0.5 - (a * 0.16666666666666666)) + (b - (b * 0.5)))) + (-1.0 - b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -35.0], 0.5, If[LessEqual[b, 3.3e+102], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(N[(a * N[(N[(0.5 - N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(b - N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -35

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-*l/ 92.7%

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

      3. associate-/r/ 92.7%

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

      4. remove-double-neg 92.7%

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

      5. unsub-neg 92.7%

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

      6. div-sub 92.7%

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

      7. *-lft-identity 92.7%

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

      8. associate-*l/ 92.7%

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

      9. lft-mult-inverse 96.4%

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

      10. sub-neg 96.4%

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

      11. distribute-frac-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -35 < b < 3.29999999999999999e102

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      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 64.7%

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

      7. *-lft-identity 64.7%

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

      8. associate-*l/ 64.7%

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

      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 79.4%

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

      1. distribute-rgt1-in 94.8%

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

      2. rec-exp 94.8%

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

      3. associate-*r/ 94.8%

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

      4. *-rgt-identity 94.8%

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

      5. +-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around 0 83.5%

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

   9. Taylor expanded in b around 0 83.5%

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

   if 3.29999999999999999e102 < 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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -35:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 - a \cdot 0.16666666666666666\right) + \left(b - b \cdot 0.5\right)\right) + \left(-1 - b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-1 + a \cdot 0.5\right)\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{2 + t\_0}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot \left(a + \frac{a}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ -1.0 (* a 0.5)))))
   (if (<= b -4.8e-178)
     (/ 1.0 (+ 2.0 t_0))
     (if (<= b 2.1e-186)
       (/ 1.0 (+ b (- 2.0 (* b (+ a (/ a b))))))
       (if (<= b 4e+102)
         (/ 1.0 (+ 2.0 (+ b t_0)))
         (/
          1.0
          (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))))

C

double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * 0.5));
	double tmp;
	if (b <= -4.8e-178) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 2.1e-186) {
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	} else if (b <= 4e+102) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	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))
    if (b <= (-4.8d-178)) then
        tmp = 1.0d0 / (2.0d0 + t_0)
    else if (b <= 2.1d-186) then
        tmp = 1.0d0 / (b + (2.0d0 - (b * (a + (a / b)))))
    else if (b <= 4d+102) then
        tmp = 1.0d0 / (2.0d0 + (b + t_0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * 0.5));
	double tmp;
	if (b <= -4.8e-178) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 2.1e-186) {
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	} else if (b <= 4e+102) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (-1.0 + (a * 0.5))
	tmp = 0
	if b <= -4.8e-178:
		tmp = 1.0 / (2.0 + t_0)
	elif b <= 2.1e-186:
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))))
	elif b <= 4e+102:
		tmp = 1.0 / (2.0 + (b + t_0))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(-1.0 + Float64(a * 0.5)))
	tmp = 0.0
	if (b <= -4.8e-178)
		tmp = Float64(1.0 / Float64(2.0 + t_0));
	elseif (b <= 2.1e-186)
		tmp = Float64(1.0 / Float64(b + Float64(2.0 - Float64(b * Float64(a + Float64(a / b))))));
	elseif (b <= 4e+102)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + t_0)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (-1.0 + (a * 0.5));
	tmp = 0.0;
	if (b <= -4.8e-178)
		tmp = 1.0 / (2.0 + t_0);
	elseif (b <= 2.1e-186)
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	elseif (b <= 4e+102)
		tmp = 1.0 / (2.0 + (b + t_0));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.8e-178], N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-186], N[(1.0 / N[(b + N[(2.0 - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+102], N[(1.0 / N[(2.0 + N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.8000000000000001e-178

   1. Initial program 95.7%

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

      1. *-lft-identity 95.7%

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

      2. associate-*l/ 95.7%

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

      3. associate-/r/ 95.7%

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

      4. remove-double-neg 95.7%

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

      5. unsub-neg 95.7%

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

      6. div-sub 81.7%

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

      7. *-lft-identity 81.7%

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

      8. associate-*l/ 81.7%

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

      9. lft-mult-inverse 97.8%

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

      10. sub-neg 97.8%

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

      11. distribute-frac-neg 97.8%

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

      12. remove-double-neg 97.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 29.9%

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

      1. distribute-rgt1-in 43.9%

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

      2. rec-exp 43.9%

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

      3. associate-*r/ 43.9%

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

      4. *-rgt-identity 43.9%

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

      5. +-commutative 43.9%

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

   7. Simplified 43.9%

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

   8. Taylor expanded in a around 0 35.0%

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

   9. Taylor expanded in b around 0 43.7%

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

   if -4.8000000000000001e-178 < b < 2.1000000000000002e-186

   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 66.0%

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

      7. *-lft-identity 66.0%

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

      8. associate-*l/ 66.0%

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

      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 78.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

      4. mul-1-neg 67.9%

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

      5. distribute-rgt-neg-in 67.9%

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

      6. distribute-neg-in 67.9%

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

      7. metadata-eval 67.9%

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

      8. unsub-neg 67.9%

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

   10. Simplified 67.9%

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

   11. Taylor expanded in b around inf 96.1%

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

       1. mul-1-neg 96.1%

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

       2. unsub-neg 96.1%

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

       3. mul-1-neg 96.1%

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

   13. Simplified 96.1%

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

   if 2.1000000000000002e-186 < b < 3.99999999999999991e102

   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 63.2%

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

      7. *-lft-identity 63.2%

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

      8. associate-*l/ 63.2%

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

      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 88.6%

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

      1. distribute-rgt1-in 88.6%

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

      2. rec-exp 88.6%

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

      3. associate-*r/ 88.6%

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

      4. *-rgt-identity 88.6%

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

      5. +-commutative 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in a around 0 66.3%

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

   9. Taylor expanded in b around 0 66.2%

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

   if 3.99999999999999991e102 < 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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot \left(a + \frac{a}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 + a \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-1 + a \cdot 0.5\right)\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{2 + t\_0}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot \left(a + \frac{a}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + \left(b + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ -1.0 (* a 0.5)))))
   (if (<= b -1.9e-179)
     (/ 1.0 (+ 2.0 t_0))
     (if (<= b 6.4e-192)
       (/ 1.0 (+ b (- 2.0 (* b (+ a (/ a b))))))
       (if (<= b 5.1e+147)
         (/ 1.0 (+ 2.0 (+ b t_0)))
         (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5))))))))))

C

double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * 0.5));
	double tmp;
	if (b <= -1.9e-179) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 6.4e-192) {
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	} else if (b <= 5.1e+147) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * ((-1.0d0) + (a * 0.5d0))
    if (b <= (-1.9d-179)) then
        tmp = 1.0d0 / (2.0d0 + t_0)
    else if (b <= 6.4d-192) then
        tmp = 1.0d0 / (b + (2.0d0 - (b * (a + (a / b)))))
    else if (b <= 5.1d+147) then
        tmp = 1.0d0 / (2.0d0 + (b + t_0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * 0.5));
	double tmp;
	if (b <= -1.9e-179) {
		tmp = 1.0 / (2.0 + t_0);
	} else if (b <= 6.4e-192) {
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	} else if (b <= 5.1e+147) {
		tmp = 1.0 / (2.0 + (b + t_0));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (-1.0 + (a * 0.5))
	tmp = 0
	if b <= -1.9e-179:
		tmp = 1.0 / (2.0 + t_0)
	elif b <= 6.4e-192:
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))))
	elif b <= 5.1e+147:
		tmp = 1.0 / (2.0 + (b + t_0))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(-1.0 + Float64(a * 0.5)))
	tmp = 0.0
	if (b <= -1.9e-179)
		tmp = Float64(1.0 / Float64(2.0 + t_0));
	elseif (b <= 6.4e-192)
		tmp = Float64(1.0 / Float64(b + Float64(2.0 - Float64(b * Float64(a + Float64(a / b))))));
	elseif (b <= 5.1e+147)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + t_0)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (-1.0 + (a * 0.5));
	tmp = 0.0;
	if (b <= -1.9e-179)
		tmp = 1.0 / (2.0 + t_0);
	elseif (b <= 6.4e-192)
		tmp = 1.0 / (b + (2.0 - (b * (a + (a / b)))));
	elseif (b <= 5.1e+147)
		tmp = 1.0 / (2.0 + (b + t_0));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.9e-179], N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.4e-192], N[(1.0 / N[(b + N[(2.0 - N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.1e+147], N[(1.0 / N[(2.0 + N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.89999999999999987e-179

   1. Initial program 95.7%

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

      1. *-lft-identity 95.7%

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

      2. associate-*l/ 95.7%

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

      3. associate-/r/ 95.7%

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

      4. remove-double-neg 95.7%

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

      5. unsub-neg 95.7%

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

      6. div-sub 81.7%

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

      7. *-lft-identity 81.7%

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

      8. associate-*l/ 81.7%

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

      9. lft-mult-inverse 97.8%

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

      10. sub-neg 97.8%

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

      11. distribute-frac-neg 97.8%

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

      12. remove-double-neg 97.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 29.9%

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

      1. distribute-rgt1-in 43.9%

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

      2. rec-exp 43.9%

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

      3. associate-*r/ 43.9%

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

      4. *-rgt-identity 43.9%

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

      5. +-commutative 43.9%

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

   7. Simplified 43.9%

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

   8. Taylor expanded in a around 0 35.0%

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

   9. Taylor expanded in b around 0 43.7%

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

   if -1.89999999999999987e-179 < b < 6.4000000000000003e-192

   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 66.0%

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

      7. *-lft-identity 66.0%

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

      8. associate-*l/ 66.0%

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

      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 78.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

      4. mul-1-neg 67.9%

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

      5. distribute-rgt-neg-in 67.9%

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

      6. distribute-neg-in 67.9%

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

      7. metadata-eval 67.9%

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

      8. unsub-neg 67.9%

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

   10. Simplified 67.9%

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

   11. Taylor expanded in b around inf 96.1%

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

       1. mul-1-neg 96.1%

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

       2. unsub-neg 96.1%

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

       3. mul-1-neg 96.1%

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

   13. Simplified 96.1%

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

   if 6.4000000000000003e-192 < b < 5.09999999999999999e147

   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 61.8%

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

      7. *-lft-identity 61.8%

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

      8. associate-*l/ 61.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 84.8%

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

      1. distribute-rgt1-in 84.8%

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

      2. rec-exp 84.8%

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

      3. associate-*r/ 84.8%

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

      4. *-rgt-identity 84.8%

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

      5. +-commutative 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in a around 0 63.5%

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

   9. Taylor expanded in b around 0 62.2%

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

   if 5.09999999999999999e147 < 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 62.2%

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

      7. *-lft-identity 62.2%

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

      8. associate-*l/ 62.2%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.6%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot \left(a + \frac{a}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 + a \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -53:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -53.0)
   0.5
   (if (<= b 6.5e+102)
     (/
      1.0
      (+ 2.0 (+ b (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666))))))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -53.0) {
		tmp = 0.5;
	} else if (b <= 6.5e+102) {
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-53.0d0)) then
        tmp = 0.5d0
    else if (b <= 6.5d+102) then
        tmp = 1.0d0 / (2.0d0 + (b + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0))))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -53.0) {
		tmp = 0.5;
	} else if (b <= 6.5e+102) {
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -53.0:
		tmp = 0.5
	elif b <= 6.5e+102:
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -53.0)
		tmp = 0.5;
	elseif (b <= 6.5e+102)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -53.0)
		tmp = 0.5;
	elseif (b <= 6.5e+102)
		tmp = 1.0 / (2.0 + (b + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -53.0], 0.5, If[LessEqual[b, 6.5e+102], N[(1.0 / N[(2.0 + N[(b + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -53

   1. Initial program 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-*l/ 92.7%

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

      3. associate-/r/ 92.7%

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

      4. remove-double-neg 92.7%

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

      5. unsub-neg 92.7%

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

      6. div-sub 92.7%

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

      7. *-lft-identity 92.7%

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

      8. associate-*l/ 92.7%

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

      9. lft-mult-inverse 96.4%

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

      10. sub-neg 96.4%

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

      11. distribute-frac-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -53 < b < 6.5000000000000004e102

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      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 64.7%

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

      7. *-lft-identity 64.7%

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

      8. associate-*l/ 64.7%

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

      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 79.4%

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

      1. distribute-rgt1-in 94.8%

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

      2. rec-exp 94.8%

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

      3. associate-*r/ 94.8%

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

      4. *-rgt-identity 94.8%

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

      5. +-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around 0 83.5%

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

   9. Taylor expanded in b around inf 84.7%

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

       1. distribute-lft-out 84.7%

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

   11. Simplified 84.7%

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

   12. Taylor expanded in b around 0 83.5%

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

   if 6.5000000000000004e102 < 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 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -53:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + \left(b + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e+146)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e+146) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8d+146) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8e+146) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8e+146:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e+146)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8e+146)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e+146], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999947e146

   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/ 98.2%

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

      4. remove-double-neg 98.2%

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

      5. unsub-neg 98.2%

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

      6. div-sub 71.2%

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

      7. *-lft-identity 71.2%

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

      8. associate-*l/ 71.2%

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

      9. lft-mult-inverse 99.1%

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

      10. sub-neg 99.1%

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

      11. distribute-frac-neg 99.1%

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

      12. remove-double-neg 99.1%

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

      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 59.9%

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

      1. distribute-rgt1-in 70.9%

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

      2. rec-exp 70.9%

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

      3. associate-*r/ 70.9%

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

      4. *-rgt-identity 70.9%

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

      5. +-commutative 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in a around 0 55.9%

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

   9. Taylor expanded in b around 0 58.9%

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

   if 7.99999999999999947e146 < 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 62.2%

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

      7. *-lft-identity 62.2%

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

      8. associate-*l/ 62.2%

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

      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 a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e-60) (/ 1.0 (- 2.0 a)) (/ 1.0 (+ b (- 2.0 (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e-60) {
		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 <= 3.2d-60) 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 <= 3.2e-60) {
		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 <= 3.2e-60:
		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 <= 3.2e-60)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(b + Float64(2.0 - Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.2e-60)
		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, 3.2e-60], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b + N[(2.0 - N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2000000000000001e-60

   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 75.8%

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

      7. *-lft-identity 75.8%

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

      8. associate-*l/ 75.8%

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

      9. lft-mult-inverse 98.9%

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

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 57.2%

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

      1. distribute-rgt1-in 70.7%

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

      2. rec-exp 70.7%

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

      3. associate-*r/ 70.7%

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

      4. *-rgt-identity 70.7%

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

      5. +-commutative 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in a around 0 48.2%

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

      1. associate-+r+ 48.2%

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

      2. +-commutative 48.2%

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

      3. associate-+l+ 48.2%

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

      4. mul-1-neg 48.2%

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

      5. distribute-rgt-neg-in 48.2%

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

      6. distribute-neg-in 48.2%

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

      7. metadata-eval 48.2%

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

      8. unsub-neg 48.2%

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

   10. Simplified 48.2%

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

   11. Taylor expanded in b around 0 52.7%

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

       1. mul-1-neg 52.7%

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

       2. unsub-neg 52.7%

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

   13. Simplified 52.7%

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

   if 3.2000000000000001e-60 < 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 56.4%

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

      7. *-lft-identity 56.4%

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

      8. associate-*l/ 56.4%

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

      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 57.8%

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

      1. distribute-rgt1-in 57.8%

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

      2. rec-exp 57.8%

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

      3. associate-*r/ 57.8%

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

      4. *-rgt-identity 57.8%

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

      5. +-commutative 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in a around 0 33.4%

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

      1. associate-+r+ 33.4%

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

      2. +-commutative 33.4%

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

      3. associate-+l+ 33.4%

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

      4. mul-1-neg 33.4%

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

      5. distribute-rgt-neg-in 33.4%

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

      6. distribute-neg-in 33.4%

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

      7. metadata-eval 33.4%

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

      8. unsub-neg 33.4%

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

   10. Simplified 33.4%

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

   11. Taylor expanded in b around inf 33.1%

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

       1. associate-*r* 33.1%

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

       2. mul-1-neg 33.1%

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

   13. Simplified 33.1%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + \left(2 - b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.2% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.75e-9) (/ 1.0 (- 2.0 a)) (/ -1.0 (* a (+ 1.0 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.75e-9) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (a * (1.0 + b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.75d-9) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-1.0d0) / (a * (1.0d0 + b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.75e-9) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (a * (1.0 + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.75e-9:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -1.0 / (a * (1.0 + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.75e-9)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-1.0 / Float64(a * Float64(1.0 + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.75e-9)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -1.0 / (a * (1.0 + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.75e-9], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7499999999999998e-9

   1. Initial program 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-*l/ 97.9%

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

      3. associate-/r/ 97.9%

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

      4. remove-double-neg 97.9%

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

      5. unsub-neg 97.9%

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

      6. div-sub 74.2%

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

      7. *-lft-identity 74.2%

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

      8. associate-*l/ 74.2%

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

      9. lft-mult-inverse 99.0%

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

      10. sub-neg 99.0%

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

      11. distribute-frac-neg 99.0%

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

      12. remove-double-neg 99.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 60.7%

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

      1. distribute-rgt1-in 73.1%

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

      2. rec-exp 73.1%

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

      3. associate-*r/ 73.1%

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

      4. *-rgt-identity 73.1%

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

      5. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in a around 0 48.5%

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

      1. associate-+r+ 48.5%

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

      2. +-commutative 48.5%

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

      3. associate-+l+ 48.5%

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

      4. mul-1-neg 48.5%

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

      5. distribute-rgt-neg-in 48.5%

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

      6. distribute-neg-in 48.5%

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

      7. metadata-eval 48.5%

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

      8. unsub-neg 48.5%

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

   10. Simplified 48.5%

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

   11. Taylor expanded in b around 0 52.4%

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

       1. mul-1-neg 52.4%

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

       2. unsub-neg 52.4%

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

   13. Simplified 52.4%

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

   if 2.7499999999999998e-9 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 56.5%

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

      7. *-lft-identity 56.5%

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

      8. associate-*l/ 56.5%

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

      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 47.0%

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

      1. distribute-rgt1-in 47.0%

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

      2. rec-exp 47.0%

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

      3. associate-*r/ 47.0%

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

      4. *-rgt-identity 47.0%

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

      5. +-commutative 47.0%

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

   7. Simplified 47.0%

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

   8. Taylor expanded in a around 0 28.7%

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

      1. associate-+r+ 28.7%

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

      2. +-commutative 28.7%

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

      3. associate-+l+ 28.7%

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

      4. mul-1-neg 28.7%

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

      5. distribute-rgt-neg-in 28.7%

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

      6. distribute-neg-in 28.7%

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

      7. metadata-eval 28.7%

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

      8. unsub-neg 28.7%

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

   10. Simplified 28.7%

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

   11. Taylor expanded in a around inf 27.4%

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

       1. +-commutative 27.4%

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

   13. Simplified 27.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(1 + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.7% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5))))))

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Python

def code(a, b):
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 57.4%

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

   1. distribute-rgt1-in 66.8%

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

   2. rec-exp 66.8%

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

   3. associate-*r/ 66.8%

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

   4. *-rgt-identity 66.8%

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

   5. +-commutative 66.8%

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

7. Simplified 66.8%

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

8. Taylor expanded in a around 0 53.9%

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

9. Taylor expanded in b around 0 54.0%

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

10. Final simplification 54.0%

    \[\leadsto \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \]
11. Add Preprocessing

Alternative 15: 40.9% 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 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 57.4%

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

   1. distribute-rgt1-in 66.8%

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

   2. rec-exp 66.8%

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

   3. associate-*r/ 66.8%

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

   4. *-rgt-identity 66.8%

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

   5. +-commutative 66.8%

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

7. Simplified 66.8%

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

8. Taylor expanded in a around 0 43.7%

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

   1. associate-+r+ 43.7%

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

   2. +-commutative 43.7%

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

   3. associate-+l+ 43.7%

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

   4. mul-1-neg 43.7%

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

   5. distribute-rgt-neg-in 43.7%

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

   6. distribute-neg-in 43.7%

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

   7. metadata-eval 43.7%

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

   8. unsub-neg 43.7%

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

10. Simplified 43.7%

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

11. Taylor expanded in b around 0 40.8%

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

    1. mul-1-neg 40.8%

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

    2. unsub-neg 40.8%

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

13. Simplified 40.8%

    \[\leadsto \color{blue}{\frac{1}{2 - a}} \]
14. Add Preprocessing

Alternative 16: 40.0% 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 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   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 80.9%

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

6. Taylor expanded in b around 0 40.0%

   \[\leadsto \color{blue}{0.5} \]
7. 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 2024089 
(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))))