Quotient of sum of exps

Percentage Accurate: 98.8% → 98.8%

Time: 8.0s

Alternatives: 9

Speedup: 1.0×

• 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: 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]

Derivation

1. Initial program 99.2%

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

Alternative 2: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -23000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -23000.0) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -23000.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-23000.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -23000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -23000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -23000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -23000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -23000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -23000

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   if -23000 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -23000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 0.16666666666666666\\ t_1 := b \cdot t\_0\\ \mathbf{if}\;b \leq 860:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+46}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{\frac{b \cdot \left(t\_0 \cdot t\_1\right) - 4}{t\_1 - 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 0.16666666666666666)) (t_1 (* b t_0)))
   (if (<= b 860.0)
     (/ (exp a) 2.0)
     (if (<= b 2.6e+46)
       (* -0.020833333333333332 (* a (* a a)))
       (if (<= b 5e+101)
         (/ 1.0 (/ (- (* b (* t_0 t_1)) 4.0) (- t_1 2.0)))
         (/ 6.0 (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * 0.16666666666666666;
	double t_1 = b * t_0;
	double tmp;
	if (b <= 860.0) {
		tmp = exp(a) / 2.0;
	} else if (b <= 2.6e+46) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 5e+101) {
		tmp = 1.0 / (((b * (t_0 * t_1)) - 4.0) / (t_1 - 2.0));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) * 0.16666666666666666d0
    t_1 = b * t_0
    if (b <= 860.0d0) then
        tmp = exp(a) / 2.0d0
    else if (b <= 2.6d+46) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    else if (b <= 5d+101) then
        tmp = 1.0d0 / (((b * (t_0 * t_1)) - 4.0d0) / (t_1 - 2.0d0))
    else
        tmp = 6.0d0 / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 0.16666666666666666;
	double t_1 = b * t_0;
	double tmp;
	if (b <= 860.0) {
		tmp = Math.exp(a) / 2.0;
	} else if (b <= 2.6e+46) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 5e+101) {
		tmp = 1.0 / (((b * (t_0 * t_1)) - 4.0) / (t_1 - 2.0));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 0.16666666666666666
	t_1 = b * t_0
	tmp = 0
	if b <= 860.0:
		tmp = math.exp(a) / 2.0
	elif b <= 2.6e+46:
		tmp = -0.020833333333333332 * (a * (a * a))
	elif b <= 5e+101:
		tmp = 1.0 / (((b * (t_0 * t_1)) - 4.0) / (t_1 - 2.0))
	else:
		tmp = 6.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 0.16666666666666666)
	t_1 = Float64(b * t_0)
	tmp = 0.0
	if (b <= 860.0)
		tmp = Float64(exp(a) / 2.0);
	elseif (b <= 2.6e+46)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	elseif (b <= 5e+101)
		tmp = Float64(1.0 / Float64(Float64(Float64(b * Float64(t_0 * t_1)) - 4.0) / Float64(t_1 - 2.0)));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 0.16666666666666666;
	t_1 = b * t_0;
	tmp = 0.0;
	if (b <= 860.0)
		tmp = exp(a) / 2.0;
	elseif (b <= 2.6e+46)
		tmp = -0.020833333333333332 * (a * (a * a));
	elseif (b <= 5e+101)
		tmp = 1.0 / (((b * (t_0 * t_1)) - 4.0) / (t_1 - 2.0));
	else
		tmp = 6.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, If[LessEqual[b, 860.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[b, 2.6e+46], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+101], N[(1.0 / N[(N[(N[(b * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 860

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 77.3%

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

   if 860 < b < 2.60000000000000013e46

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 3.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(0 - \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)\right) - \frac{1}{48}\right)\right)\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \color{blue}{\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{48}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \color{blue}{\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{5}{48} + \frac{1}{8}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \frac{11}{48}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \left(\frac{1}{24} + \color{blue}{\frac{3}{16}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \color{blue}{\frac{3}{16} \cdot b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \frac{3}{16} \cdot b\right)\right)\right)\right)}\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{1}{24} + \frac{3}{16}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 89.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 2.60000000000000013e46 < b < 4.99999999999999989e101

   1. Initial program 91.7%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 7.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 7.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{6} \cdot {b}^{2}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 7.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   11. Simplified 7.0%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) + \color{blue}{2}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right) - 2 \cdot 2}{\color{blue}{b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right) - 2 \cdot 2\right), \color{blue}{\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\color{blue}{b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)} - 2\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(\left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\frac{1}{6} \cdot \left(b \cdot b\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\left(b \cdot b\right) \cdot \frac{1}{6}\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \frac{1}{6}\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \left(\left(b \cdot b\right) \cdot \frac{1}{6}\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(b \cdot b\right), \frac{1}{6}\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right)\right)\right)\right), \left(2 \cdot 2\right)\right), \left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right) - 2\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right)\right)\right)\right), 4\right), \left(b \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)} - 2\right)\right)\right) \]

       16. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{6}\right)\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\left(b \cdot \left(\frac{1}{6} \cdot \left(b \cdot b\right)\right)\right), \color{blue}{2}\right)\right)\right) \]

   13. Applied egg-rr 92.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot \left(\left(\left(b \cdot b\right) \cdot 0.16666666666666666\right) \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)\right)\right) - 4}{b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right) - 2}}} \]

   if 4.99999999999999989e101 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.4% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -620:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(0.16666666666666666 + \frac{0.5 + \frac{1}{b}}{b}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{\frac{6}{b \cdot b}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -620.0)
   (/ 1.0 (* b (* b (* b (+ 0.16666666666666666 (/ (+ 0.5 (/ 1.0 b)) b))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (/ 1.0 (/ 6.0 (* b b)))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -620.0) {
		tmp = 1.0 / (b * (b * (b * (0.16666666666666666 + ((0.5 + (1.0 / b)) / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (1.0 / (6.0 / (b * b))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-620.0d0)) then
        tmp = 1.0d0 / (b * (b * (b * (0.16666666666666666d0 + ((0.5d0 + (1.0d0 / b)) / b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (1.0d0 / (6.0d0 / (b * b))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -620.0) {
		tmp = 1.0 / (b * (b * (b * (0.16666666666666666 + ((0.5 + (1.0 / b)) / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (1.0 / (6.0 / (b * b))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -620.0:
		tmp = 1.0 / (b * (b * (b * (0.16666666666666666 + ((0.5 + (1.0 / b)) / b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (1.0 / (6.0 / (b * b))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -620.0)
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * Float64(0.16666666666666666 + Float64(Float64(0.5 + Float64(1.0 / b)) / b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(1.0 / Float64(6.0 / Float64(b * b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -620.0)
		tmp = 1.0 / (b * (b * (b * (0.16666666666666666 + ((0.5 + (1.0 / b)) / b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (1.0 / (6.0 / (b * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -620.0], N[(1.0 / N[(b * N[(b * N[(b * N[(0.16666666666666666 + N[(N[(0.5 + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(1.0 / N[(6.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -620

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 29.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 29.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 24.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 24.9%

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

   9. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left({b}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left({b}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({b}^{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(b \cdot {b}^{2}\right) \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2} \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} - \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) + -1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)\right)\right)\right)\right) \]

       12. distribute-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)}\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + \frac{1}{b}}{b}}\right)\right)\right)\right)\right)\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. remove-double-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} + \frac{\frac{1}{2} + \frac{1}{b}}{\color{blue}{b}}\right)\right)\right)\right) \]

       17. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{\frac{1}{2} + \frac{1}{b}}{b}\right)}\right)\right)\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{b}\right), \color{blue}{b}\right)\right)\right)\right)\right) \]

   11. Simplified 28.6%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{6} + \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)}\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \left(\left(b \cdot \left(\frac{1}{6} + \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right) \cdot \color{blue}{b}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(b \cdot \left(\frac{1}{6} + \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right), \color{blue}{b}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{1}{6} + \frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right), b\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{\frac{1}{2} + \frac{1}{b}}{b}\right)\right)\right), b\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{b}\right), b\right)\right)\right), b\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{b}\right)\right), b\right)\right)\right), b\right)\right)\right) \]

       8. /-lowering-/.f64 64.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(1, b\right)\right), b\right)\right)\right), b\right)\right)\right) \]

   13. Applied egg-rr 64.8%

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

   if -620 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 67.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 67.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{1}{2} - b \cdot \frac{1}{6}}{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}}}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} - b \cdot \frac{1}{6}}{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}\right)}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - b \cdot \frac{1}{6}\right), \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{6} \cdot b\right), \left(\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}\right) \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot b\right), \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)} \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot b\right)\right), \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)} \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{6}\right)\right), b\right)\right), \left(\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}\right) \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \left(\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{\left(b \cdot \frac{1}{6}\right)} \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{4} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

      15. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval 67.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{6}, b\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.1%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\frac{1}{\frac{0.5 + -0.16666666666666666 \cdot b}{\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b}}}\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{6}{{b}^{2}}\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 67.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 67.1%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \frac{1}{\color{blue}{\frac{6}{b \cdot b}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -620:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(0.16666666666666666 + \frac{0.5 + \frac{1}{b}}{b}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{\frac{6}{b \cdot b}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot -0.020833333333333332\right)\\ \mathbf{if}\;b \leq 350:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + t\_0\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+101}:\\ \;\;\;\;a \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a -0.020833333333333332))))
   (if (<= b 350.0)
     (+ 0.5 (* a (+ 0.25 t_0)))
     (if (<= b 3.6e+101) (* a t_0) (/ 6.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double t_0 = a * (a * -0.020833333333333332);
	double tmp;
	if (b <= 350.0) {
		tmp = 0.5 + (a * (0.25 + t_0));
	} else if (b <= 3.6e+101) {
		tmp = a * t_0;
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * (-0.020833333333333332d0))
    if (b <= 350.0d0) then
        tmp = 0.5d0 + (a * (0.25d0 + t_0))
    else if (b <= 3.6d+101) then
        tmp = a * t_0
    else
        tmp = 6.0d0 / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * -0.020833333333333332);
	double tmp;
	if (b <= 350.0) {
		tmp = 0.5 + (a * (0.25 + t_0));
	} else if (b <= 3.6e+101) {
		tmp = a * t_0;
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * -0.020833333333333332)
	tmp = 0
	if b <= 350.0:
		tmp = 0.5 + (a * (0.25 + t_0))
	elif b <= 3.6e+101:
		tmp = a * t_0
	else:
		tmp = 6.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * -0.020833333333333332))
	tmp = 0.0
	if (b <= 350.0)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + t_0)));
	elseif (b <= 3.6e+101)
		tmp = Float64(a * t_0);
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * -0.020833333333333332);
	tmp = 0.0;
	if (b <= 350.0)
		tmp = 0.5 + (a * (0.25 + t_0));
	elseif (b <= 3.6e+101)
		tmp = a * t_0;
	else
		tmp = 6.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 350.0], N[(0.5 + N[(a * N[(0.25 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+101], N[(a * t$95$0), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 350

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(0 - \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)\right) - \frac{1}{48}\right)\right)\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \color{blue}{\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{48}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \color{blue}{\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{5}{48} + \frac{1}{8}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \frac{11}{48}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \left(\frac{1}{24} + \color{blue}{\frac{3}{16}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \color{blue}{\frac{3}{16} \cdot b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \frac{3}{16} \cdot b\right)\right)\right)\right)}\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{1}{24} + \frac{3}{16}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 55.9%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 350 < b < 3.60000000000000029e101

   1. Initial program 95.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 12.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(0 - \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)\right) - \frac{1}{48}\right)\right)\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \color{blue}{\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{48}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \color{blue}{\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{5}{48} + \frac{1}{8}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \frac{11}{48}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \left(\frac{1}{24} + \color{blue}{\frac{3}{16}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \color{blue}{\frac{3}{16} \cdot b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \frac{3}{16} \cdot b\right)\right)\right)\right)}\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{1}{24} + \frac{3}{16}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 59.8%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{48} \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{48} \cdot a\right) \cdot a\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{48} \cdot a\right), a\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \frac{-1}{48}\right), a\right), a\right) \]

       6. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{48}\right), a\right), a\right) \]

   13. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{\left(\left(a \cdot -0.020833333333333332\right) \cdot a\right) \cdot a} \]

   if 3.60000000000000029e101 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 98.3%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 98.3%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 98.3%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 350:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 420:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 420.0)
   (+ 0.5 (* a 0.25))
   (if (<= b 3.6e+101)
     (* a (* a (* a -0.020833333333333332)))
     (/ 6.0 (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 420.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 3.6e+101) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 420.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 3.6d+101) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 6.0d0 / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 420.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 3.6e+101) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 420.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 3.6e+101:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 6.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 420.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 3.6e+101)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 420.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 3.6e+101)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 6.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 420.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+101], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 420

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

      17. exp-lowering-exp.f64 77.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 77.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-lowering-*.f64 55.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

   8. Simplified 55.9%

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

   if 420 < b < 3.60000000000000029e101

   1. Initial program 95.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 12.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(0 - \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)\right) - \frac{1}{48}\right)\right)\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \color{blue}{\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{48}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \color{blue}{\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{5}{48} + \frac{1}{8}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \frac{11}{48}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \left(\frac{1}{24} + \color{blue}{\frac{3}{16}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \color{blue}{\frac{3}{16} \cdot b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \frac{3}{16} \cdot b\right)\right)\right)\right)}\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{1}{24} + \frac{3}{16}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 59.8%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{48} \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{48} \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-1}{48} \cdot a\right) \cdot a\right), \color{blue}{a}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{48} \cdot a\right), a\right), a\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \frac{-1}{48}\right), a\right), a\right) \]

       6. *-lowering-*.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{48}\right), a\right), a\right) \]

   13. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{\left(\left(a \cdot -0.020833333333333332\right) \cdot a\right) \cdot a} \]

   if 3.60000000000000029e101 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 98.3%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 98.3%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 98.3%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 420:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 370:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 370.0) (+ 0.5 (* a 0.25)) (* -0.020833333333333332 (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 370.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -0.020833333333333332 * (a * (a * 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 <= 370.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 370.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -0.020833333333333332 * (a * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 370.0:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -0.020833333333333332 * (a * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 370.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 370.0)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -0.020833333333333332 * (a * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 370.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 370

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

      17. exp-lowering-exp.f64 77.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 77.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-lowering-*.f64 55.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

   8. Simplified 55.9%

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

   if 370 < b

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 23.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(0 - \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)\right) - \frac{1}{48}\right)\right)\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \color{blue}{\left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{48}\right)}\right)\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \color{blue}{\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{5}{48} + \frac{1}{8}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \frac{11}{48}\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \left(\frac{1}{24} + \color{blue}{\frac{3}{16}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \color{blue}{\frac{3}{16} \cdot b}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + \left(\frac{1}{24} \cdot b + \frac{3}{16} \cdot b\right)\right)\right)\right)}\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \left(a \cdot \left(\left(\frac{5}{48} \cdot b + \frac{1}{8} \cdot b\right) - \left(\frac{1}{48} + b \cdot \color{blue}{\left(\frac{1}{24} + \frac{3}{16}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.8%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 51.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 51.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 370:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.5% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

   5. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

   15. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

   17. exp-lowering-exp.f64 83.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

5. Simplified 83.6%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

   2. *-lowering-*.f64 41.0%

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

8. Simplified 41.0%

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

9. Final simplification 41.0%

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

Alternative 9: 40.3% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 83.2%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

5. Simplified 83.2%

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

6. Taylor expanded in b around 0

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

8. Simplified 40.7%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024150 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))