Quotient of sum of exps

Percentage Accurate: 98.8% → 98.8%

Time: 8.6s

Alternatives: 16

Speedup: 1.3×

• 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 99.6%

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

Alternative 2: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(1 + a \cdot 0.5\right)\\ \mathbf{if}\;e^{a} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{\left(e^{b} + 1\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ 1.0 (* a 0.5)))))
   (if (<= (exp a) 1.0)
     (/ (exp a) (+ (exp a) 1.0))
     (/ (+ 1.0 t_0) (+ (+ (exp b) 1.0) t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (1.0 + (a * 0.5));
	double tmp;
	if (exp(a) <= 1.0) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = (1.0 + t_0) / ((exp(b) + 1.0) + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 + (a * 0.5d0))
    if (exp(a) <= 1.0d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = (1.0d0 + t_0) / ((exp(b) + 1.0d0) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (1.0 + (a * 0.5));
	double tmp;
	if (Math.exp(a) <= 1.0) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = (1.0 + t_0) / ((Math.exp(b) + 1.0) + t_0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (1.0 + (a * 0.5))
	tmp = 0
	if math.exp(a) <= 1.0:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = (1.0 + t_0) / ((math.exp(b) + 1.0) + t_0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(1.0 + Float64(a * 0.5)))
	tmp = 0.0
	if (exp(a) <= 1.0)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(1.0 + t_0) / Float64(Float64(exp(b) + 1.0) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (1.0 + (a * 0.5));
	tmp = 0.0;
	if (exp(a) <= 1.0)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = (1.0 + t_0) / ((exp(b) + 1.0) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1

   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 69.0%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 90.5%

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

   8. Simplified 90.5%

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

4. Final simplification 69.8%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(1 + a \cdot 0.5\right)\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{\left(e^{b} + 1\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ 1.0 (* a 0.5)))))
   (if (<= (exp a) 0.0)
     (/ (exp a) 2.0)
     (/ (+ 1.0 t_0) (+ (+ (exp b) 1.0) t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (1.0 + (a * 0.5));
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = (1.0 + t_0) / ((exp(b) + 1.0) + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 + (a * 0.5d0))
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / ((exp(b) + 1.0d0) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (1.0 + (a * 0.5));
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = (1.0 + t_0) / ((Math.exp(b) + 1.0) + t_0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (1.0 + (a * 0.5))
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = (1.0 + t_0) / ((math.exp(b) + 1.0) + t_0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(1.0 + Float64(a * 0.5)))
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / Float64(Float64(exp(b) + 1.0) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (1.0 + (a * 0.5));
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (1.0 + t_0) / ((exp(b) + 1.0) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. 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 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a \cdot \left(1 + a \cdot 0.5\right)}{\left(e^{b} + 1\right) + a \cdot \left(1 + a \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{\left(e^{b} + 1\right) + a \cdot \left(1 + a \cdot 0.5\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return Math.exp(a) / ((Math.exp(b) + 1.0) + (a * (1.0 + (a * 0.5))));
}

Python

def code(a, b):
	return math.exp(a) / ((math.exp(b) + 1.0) + (a * (1.0 + (a * 0.5))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

   \[\leadsto \frac{e^{a}}{\left(e^{b} + 1\right) + a \cdot \left(1 + a \cdot 0.5\right)} \]
7. Add Preprocessing

Alternative 5: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999998) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999998) {
		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 (exp(a) <= 0.9999998d0) 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 (Math.exp(a) <= 0.9999998) {
		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 math.exp(a) <= 0.9999998:
		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 (exp(a) <= 0.9999998)
		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 (exp(a) <= 0.9999998)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999998], 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 (exp.f64 a) < 0.999999799999999994

   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 98.3%

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

   if 0.999999799999999994 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   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 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -15.0)
   1.0
   (if (<= b 1e+103)
     (/ (exp a) 2.0)
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -15.0) {
		tmp = 1.0;
	} else if (b <= 1e+103) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-15.0d0)) then
        tmp = 1.0d0
    else if (b <= 1d+103) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -15.0) {
		tmp = 1.0;
	} else if (b <= 1e+103) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -15.0:
		tmp = 1.0
	elif b <= 1e+103:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -15.0)
		tmp = 1.0;
	elseif (b <= 1e+103)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -15.0)
		tmp = 1.0;
	elseif (b <= 1e+103)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -15.0], 1.0, If[LessEqual[b, 1e+103], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -15

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -15 < b < 1e103

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 92.7%

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

   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 91.0%

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

   if 1e103 < 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%

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

Alternative 7: 76.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+97}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13)
   1.0
   (if (<= b 2.9e-15)
     (+
      0.5
      (*
       a
       (+
        0.25
        (*
         (* a a)
         (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
     (if (<= b 6e+97)
       (* 0.0020833333333333333 (* a (* a (* a (* a a)))))
       (/
        1.0
        (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 6e+97) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 2.9d-15) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 6d+97) then
        tmp = 0.0020833333333333333d0 * (a * (a * (a * (a * a))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 6e+97) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 2.9e-15:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 6e+97:
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 6e+97)
		tmp = Float64(0.0020833333333333333 * Float64(a * Float64(a * Float64(a * Float64(a * a)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 6e+97)
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 2.9e-15], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+97], N[(0.0020833333333333333 * N[(a * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 2.90000000000000019e-15

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 67.5%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 2.90000000000000019e-15 < b < 5.9999999999999997e97

   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 40.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{4} + \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \color{blue}{a}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)} \cdot a\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a}{\color{blue}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a\right), \color{blue}{\left(\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}\right)\right) \]

   10. Applied egg-rr 2.1%

       \[\leadsto 0.5 + \color{blue}{\frac{\left(0.0625 - \left(a \cdot a\right) \cdot \left(\left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right) \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\right)\right) \cdot a}{0.25 - a \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{\frac{1}{480} \cdot {a}^{5}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \color{blue}{\left({a}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{4} \cdot \color{blue}{a}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left(a \cdot \color{blue}{{a}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{3}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 56.5%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   13. Simplified 56.5%

       \[\leadsto \color{blue}{0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)} \]

   if 5.9999999999999997e97 < 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 98.4%

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

   8. Simplified 98.4%

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

Alternative 8: 76.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+97}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13)
   1.0
   (if (<= b 2.9e-15)
     (+ 0.5 (* a (+ 0.25 (* (* a a) -0.020833333333333332))))
     (if (<= b 6e+97)
       (* 0.0020833333333333333 (* a (* a (* a (* a a)))))
       (/
        1.0
        (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	} else if (b <= 6e+97) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 2.9d-15) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * (-0.020833333333333332d0))))
    else if (b <= 6d+97) then
        tmp = 0.0020833333333333333d0 * (a * (a * (a * (a * a))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	} else if (b <= 6e+97) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 2.9e-15:
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)))
	elif b <= 6e+97:
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * -0.020833333333333332))));
	elseif (b <= 6e+97)
		tmp = Float64(0.0020833333333333333 * Float64(a * Float64(a * Float64(a * Float64(a * a)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	elseif (b <= 6e+97)
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 2.9e-15], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+97], N[(0.0020833333333333333 * N[(a * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 2.90000000000000019e-15

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.2%

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

   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. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 67.3%

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

   8. Simplified 67.3%

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

   if 2.90000000000000019e-15 < b < 5.9999999999999997e97

   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 40.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{4} + \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \color{blue}{a}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)} \cdot a\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a}{\color{blue}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a\right), \color{blue}{\left(\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}\right)\right) \]

   10. Applied egg-rr 2.1%

       \[\leadsto 0.5 + \color{blue}{\frac{\left(0.0625 - \left(a \cdot a\right) \cdot \left(\left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right) \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\right)\right) \cdot a}{0.25 - a \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{\frac{1}{480} \cdot {a}^{5}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \color{blue}{\left({a}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{4} \cdot \color{blue}{a}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left(a \cdot \color{blue}{{a}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{3}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 56.5%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   13. Simplified 56.5%

       \[\leadsto \color{blue}{0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)} \]

   if 5.9999999999999997e97 < 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 98.4%

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

   8. Simplified 98.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+97}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13)
   1.0
   (if (<= b 2.9e-15)
     (+ 0.5 (* a (+ 0.25 (* (* a a) -0.020833333333333332))))
     (if (<= b 1.3e+150)
       (* 0.0020833333333333333 (* a (* a (* a (* a a)))))
       (/ 1.0 (* b (+ 1.0 (* b 0.5))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	} else if (b <= 1.3e+150) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 2.9d-15) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * (-0.020833333333333332d0))))
    else if (b <= 1.3d+150) then
        tmp = 0.0020833333333333333d0 * (a * (a * (a * (a * a))))
    else
        tmp = 1.0d0 / (b * (1.0d0 + (b * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	} else if (b <= 1.3e+150) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 2.9e-15:
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)))
	elif b <= 1.3e+150:
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))))
	else:
		tmp = 1.0 / (b * (1.0 + (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * -0.020833333333333332))));
	elseif (b <= 1.3e+150)
		tmp = Float64(0.0020833333333333333 * Float64(a * Float64(a * Float64(a * Float64(a * a)))));
	else
		tmp = Float64(1.0 / Float64(b * Float64(1.0 + Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = 0.5 + (a * (0.25 + ((a * a) * -0.020833333333333332)));
	elseif (b <= 1.3e+150)
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	else
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 2.9e-15], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+150], N[(0.0020833333333333333 * N[(a * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 2.90000000000000019e-15

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.2%

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

   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. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 67.3%

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

   8. Simplified 67.3%

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

   if 2.90000000000000019e-15 < b < 1.30000000000000003e150

   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 45.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 2.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.5%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{4} + \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \color{blue}{a}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)} \cdot a\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a}{\color{blue}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a\right), \color{blue}{\left(\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}\right)\right) \]

   10. Applied egg-rr 2.0%

       \[\leadsto 0.5 + \color{blue}{\frac{\left(0.0625 - \left(a \cdot a\right) \cdot \left(\left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right) \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\right)\right) \cdot a}{0.25 - a \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{\frac{1}{480} \cdot {a}^{5}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \color{blue}{\left({a}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{4} \cdot \color{blue}{a}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left(a \cdot \color{blue}{{a}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{3}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 51.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   13. Simplified 51.0%

       \[\leadsto \color{blue}{0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)} \]

   if 1.30000000000000003e150 < 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

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

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

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

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

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

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

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

      4. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lft-mult-inverse N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13)
   1.0
   (if (<= b 360.0)
     (+ 0.5 (* a 0.25))
     (if (<= b 1.3e+150)
       (* 0.0020833333333333333 (* a (* a (* a (* a a)))))
       (/ 1.0 (* b (+ 1.0 (* b 0.5))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 360.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.3e+150) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 360.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.3d+150) then
        tmp = 0.0020833333333333333d0 * (a * (a * (a * (a * a))))
    else
        tmp = 1.0d0 / (b * (1.0d0 + (b * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 360.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.3e+150) {
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 360.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.3e+150:
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))))
	else:
		tmp = 1.0 / (b * (1.0 + (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 360.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.3e+150)
		tmp = Float64(0.0020833333333333333 * Float64(a * Float64(a * Float64(a * Float64(a * a)))));
	else
		tmp = Float64(1.0 / Float64(b * Float64(1.0 + Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 360.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.3e+150)
		tmp = 0.0020833333333333333 * (a * (a * (a * (a * a))));
	else
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 360.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+150], N[(0.0020833333333333333 * N[(a * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 360

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 66.8%

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

   5. Simplified 66.8%

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

   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 66.8%

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

   8. Simplified 66.8%

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

   if 360 < b < 1.30000000000000003e150

   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 43.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\frac{1}{4} + \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \color{blue}{a}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)} \cdot a\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a}{\color{blue}{\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{4} \cdot \frac{1}{4} - \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)\right) \cdot a\right), \color{blue}{\left(\frac{1}{4} - \left(a \cdot a\right) \cdot \left(\frac{-1}{48} + \left(a \cdot a\right) \cdot \frac{1}{480}\right)\right)}\right)\right) \]

   10. Applied egg-rr 2.1%

       \[\leadsto 0.5 + \color{blue}{\frac{\left(0.0625 - \left(a \cdot a\right) \cdot \left(\left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right) \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\right)\right) \cdot a}{0.25 - a \cdot \left(a \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)}} \]

   11. Taylor expanded in a around inf

       \[\leadsto \color{blue}{\frac{1}{480} \cdot {a}^{5}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \color{blue}{\left({a}^{5}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{\left(4 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left({a}^{4} \cdot \color{blue}{a}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \left(a \cdot \color{blue}{{a}^{4}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{4}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left({a}^{2} \cdot \color{blue}{{a}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(\left(a \cdot a\right) \cdot {\color{blue}{a}}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot \left(a \cdot \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{3}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot {a}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 52.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{480}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right) \]

   13. Simplified 52.8%

       \[\leadsto \color{blue}{0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)} \]

   if 1.30000000000000003e150 < 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

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

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

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

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

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

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

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

      4. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lft-mult-inverse N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 360:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;0.0020833333333333333 \cdot \left(a \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13)
   1.0
   (if (<= b 2.9e-15) (+ 0.5 (* a 0.25)) (/ 1.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 2.9d-15) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / (b * (1.0d0 + (b * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 2.9e-15:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (b * (1.0 + (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(b * Float64(1.0 + Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (b * (1.0 + (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 2.9e-15], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 2.90000000000000019e-15

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 67.3%

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

   5. Simplified 67.3%

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

   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 67.3%

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

   8. Simplified 67.3%

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

   if 2.90000000000000019e-15 < 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 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 60.9%

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

   8. Simplified 60.9%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lft-mult-inverse N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 60.9%

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

   11. Simplified 60.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.7% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -8e-13) 1.0 (if (<= b 2.9e-15) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (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 (b <= (-8d-13)) then
        tmp = 1.0d0
    else if (b <= 2.9d-15) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else if (b <= 2.9e-15) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	elif b <= 2.9e-15:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	elseif (b <= 2.9e-15)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, If[LessEqual[b, 2.9e-15], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b < 2.90000000000000019e-15

   1. Initial program 99.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 67.3%

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

   5. Simplified 67.3%

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

   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 67.3%

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

   8. Simplified 67.3%

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

   if 2.90000000000000019e-15 < 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 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 60.9%

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

   8. Simplified 60.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 60.9%

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

   11. Simplified 60.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.4% accurate, 30.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1 < b

   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 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 45.2%

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

   8. Simplified 45.2%

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

Alternative 14: 54.5% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -8e-13) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-8d-13)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -8e-13) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -8e-13:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -8e-13)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -8e-13)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -8e-13], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.0000000000000002e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 97.6%

       \[\leadsto \color{blue}{1} \]

   if -8.0000000000000002e-13 < b

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right) + \frac{1}{1 + e^{b}}} \]
   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 78.2%

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

   5. Simplified 78.2%

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

   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 44.9%

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

   8. Simplified 44.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.5% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.1000000000000001 < b

   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 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 44.3%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 39.9% 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 80.8%

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

5. Simplified 80.8%

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

6. Taylor expanded in b around 0

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

8. Simplified 40.4%

   \[\leadsto \color{blue}{0.5} \]
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 2024138 
(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))))