Quotient of sum of exps

Percentage Accurate: 98.9% → 99.2%

Time: 8.7s

Alternatives: 17

Speedup: 1.3×

• could not determine a ground truth

  1. a = 1.061871156409709e+167
  2. b = -1.5777327931363248e+157

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{t\_0 + \left(e^{b} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ 1.0 (* a (+ 0.5 (* a 0.16666666666666666)))))))
   (if (<= (exp a) 0.9999)
     (/ (exp a) (+ (exp a) 1.0))
     (/ (+ 1.0 t_0) (+ t_0 (+ (exp b) 1.0))))))

C

double code(double a, double b) {
	double t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	double tmp;
	if (exp(a) <= 0.9999) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = (1.0 + t_0) / (t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 + (a * (0.5d0 + (a * 0.16666666666666666d0))))
    if (exp(a) <= 0.9999d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = (1.0d0 + t_0) / (t_0 + (exp(b) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	double tmp;
	if (Math.exp(a) <= 0.9999) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = (1.0 + t_0) / (t_0 + (Math.exp(b) + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))
	tmp = 0
	if math.exp(a) <= 0.9999:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = (1.0 + t_0) / (t_0 + (math.exp(b) + 1.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(a * 0.16666666666666666)))))
	tmp = 0.0
	if (exp(a) <= 0.9999)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(1.0 + t_0) / Float64(t_0 + Float64(exp(b) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	tmp = 0.0;
	if (exp(a) <= 0.9999)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = (1.0 + t_0) / (t_0 + (exp(b) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(1.0 + N[(a * N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(t$95$0 + N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99990000000000001

   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%

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

   if 0.99990000000000001 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)}{a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right) + \left(e^{b} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{t\_0 + \left(e^{b} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ 1.0 (* a (+ 0.5 (* a 0.16666666666666666)))))))
   (if (<= (exp a) 0.0)
     (/ (exp a) 2.0)
     (/ (+ 1.0 t_0) (+ t_0 (+ (exp b) 1.0))))))

C

double code(double a, double b) {
	double t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = (1.0 + t_0) / (t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (1.0d0 + (a * (0.5d0 + (a * 0.16666666666666666d0))))
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / (t_0 + (exp(b) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = (1.0 + t_0) / (t_0 + (Math.exp(b) + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = (1.0 + t_0) / (t_0 + (math.exp(b) + 1.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(a * 0.16666666666666666)))))
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / Float64(t_0 + Float64(exp(b) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))));
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (1.0 + t_0) / (t_0 + (exp(b) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(1.0 + N[(a * N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(t$95$0 + N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $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 98.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)}{a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right) + \left(e^{b} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99996:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99996) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99996) {
		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) <= 0.99996d0) 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) <= 0.99996) {
		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) <= 0.99996:
		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) <= 0.99996)
		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) <= 0.99996)
		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], 0.99996], 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) < 0.99995999999999996

   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 0.99995999999999996 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99996:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\\ \mathbf{if}\;b \leq -17.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 + \left(b \cdot \left(1 + t\_0\right)\right) \cdot \left(b \cdot \left(-1 - t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (+ 0.5 (* b 0.16666666666666666)))))
   (if (<= b -17.5)
     1.0
     (if (<= b 4.8e+45)
       (/ (exp a) 2.0)
       (if (<= b 5e+102)
         (/
          (+ 2.0 (* b (+ -1.0 (* b (+ -0.5 (* b -0.16666666666666666))))))
          (+ 4.0 (* (* b (+ 1.0 t_0)) (* b (- -1.0 t_0)))))
         (/ -4.0 (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = b * (0.5 + (b * 0.16666666666666666));
	double tmp;
	if (b <= -17.5) {
		tmp = 1.0;
	} else if (b <= 4.8e+45) {
		tmp = exp(a) / 2.0;
	} else if (b <= 5e+102) {
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (0.5d0 + (b * 0.16666666666666666d0))
    if (b <= (-17.5d0)) then
        tmp = 1.0d0
    else if (b <= 4.8d+45) then
        tmp = exp(a) / 2.0d0
    else if (b <= 5d+102) then
        tmp = (2.0d0 + (b * ((-1.0d0) + (b * ((-0.5d0) + (b * (-0.16666666666666666d0))))))) / (4.0d0 + ((b * (1.0d0 + t_0)) * (b * ((-1.0d0) - t_0))))
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (0.5 + (b * 0.16666666666666666));
	double tmp;
	if (b <= -17.5) {
		tmp = 1.0;
	} else if (b <= 4.8e+45) {
		tmp = Math.exp(a) / 2.0;
	} else if (b <= 5e+102) {
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (0.5 + (b * 0.16666666666666666))
	tmp = 0
	if b <= -17.5:
		tmp = 1.0
	elif b <= 4.8e+45:
		tmp = math.exp(a) / 2.0
	elif b <= 5e+102:
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))))
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))
	tmp = 0.0
	if (b <= -17.5)
		tmp = 1.0;
	elseif (b <= 4.8e+45)
		tmp = Float64(exp(a) / 2.0);
	elseif (b <= 5e+102)
		tmp = Float64(Float64(2.0 + Float64(b * Float64(-1.0 + Float64(b * Float64(-0.5 + Float64(b * -0.16666666666666666)))))) / Float64(4.0 + Float64(Float64(b * Float64(1.0 + t_0)) * Float64(b * Float64(-1.0 - t_0)))));
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (0.5 + (b * 0.16666666666666666));
	tmp = 0.0;
	if (b <= -17.5)
		tmp = 1.0;
	elseif (b <= 4.8e+45)
		tmp = exp(a) / 2.0;
	elseif (b <= 5e+102)
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -17.5], 1.0, If[LessEqual[b, 4.8e+45], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[b, 5e+102], N[(N[(2.0 + N[(b * N[(-1.0 + N[(b * N[(-0.5 + N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(N[(b * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -17.5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -17.5 < b < 4.79999999999999979e45

   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 95.2%

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

   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 92.2%

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

   if 4.79999999999999979e45 < b < 5e102

   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 5.6%

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

   8. Simplified 5.6%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)}{\color{blue}{2 - b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{2 - b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)}{\color{blue}{2 \cdot 2 - \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)}} \]

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

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

   10. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 - \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)}} \]

   if 5e102 < b

   1. Initial program 97.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 68.6%

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

   8. Simplified 68.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

       10. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 68.6%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   14. Simplified 100.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -17.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 + \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \cdot \left(b \cdot \left(-1 - b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + b \cdot 0.16666666666666666\\ t_1 := b \cdot t\_0\\ \mathbf{if}\;b \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 + \left(b \cdot t\_1\right) \cdot \left(b \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}{1 + t\_1 \cdot \left(-1 + t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* b 0.16666666666666666))) (t_1 (* b t_0)))
   (if (<= b -1.55)
     1.0
     (if (<= b 2.8e+77)
       (/
        1.0
        (+
         2.0
         (/
          (* b (+ 1.0 (* (* b t_1) (* b (* t_0 t_0)))))
          (+ 1.0 (* t_1 (+ -1.0 t_1))))))
       (/ -4.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double t_0 = 0.5 + (b * 0.16666666666666666);
	double t_1 = b * t_0;
	double tmp;
	if (b <= -1.55) {
		tmp = 1.0;
	} else if (b <= 2.8e+77) {
		tmp = 1.0 / (2.0 + ((b * (1.0 + ((b * t_1) * (b * (t_0 * t_0))))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 + (b * 0.16666666666666666d0)
    t_1 = b * t_0
    if (b <= (-1.55d0)) then
        tmp = 1.0d0
    else if (b <= 2.8d+77) then
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 + ((b * t_1) * (b * (t_0 * t_0))))) / (1.0d0 + (t_1 * ((-1.0d0) + t_1)))))
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 0.5 + (b * 0.16666666666666666);
	double t_1 = b * t_0;
	double tmp;
	if (b <= -1.55) {
		tmp = 1.0;
	} else if (b <= 2.8e+77) {
		tmp = 1.0 / (2.0 + ((b * (1.0 + ((b * t_1) * (b * (t_0 * t_0))))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 + (b * 0.16666666666666666)
	t_1 = b * t_0
	tmp = 0
	if b <= -1.55:
		tmp = 1.0
	elif b <= 2.8e+77:
		tmp = 1.0 / (2.0 + ((b * (1.0 + ((b * t_1) * (b * (t_0 * t_0))))) / (1.0 + (t_1 * (-1.0 + t_1)))))
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 + Float64(b * 0.16666666666666666))
	t_1 = Float64(b * t_0)
	tmp = 0.0
	if (b <= -1.55)
		tmp = 1.0;
	elseif (b <= 2.8e+77)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(b * Float64(1.0 + Float64(Float64(b * t_1) * Float64(b * Float64(t_0 * t_0))))) / Float64(1.0 + Float64(t_1 * Float64(-1.0 + t_1))))));
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 + (b * 0.16666666666666666);
	t_1 = b * t_0;
	tmp = 0.0;
	if (b <= -1.55)
		tmp = 1.0;
	elseif (b <= 2.8e+77)
		tmp = 1.0 / (2.0 + ((b * (1.0 + ((b * t_1) * (b * (t_0 * t_0))))) / (1.0 + (t_1 * (-1.0 + t_1)))));
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, If[LessEqual[b, -1.55], 1.0, If[LessEqual[b, 2.8e+77], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 + N[(N[(b * t$95$1), $MachinePrecision] * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -1.55000000000000004 < b < 2.8e77

   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 66.6%

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

   5. Simplified 66.6%

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

   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 57.9%

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

   8. Simplified 57.9%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 62.5%

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

   if 2.8e77 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   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 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

       10. *-lowering-*.f64 64.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   14. Simplified 93.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 + \left(b \cdot \left(b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \cdot \left(b \cdot \left(\left(0.5 + b \cdot 0.16666666666666666\right) \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\right)}{1 + \left(b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right) \cdot \left(-1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\\ \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 + \left(b \cdot \left(1 + t\_0\right)\right) \cdot \left(b \cdot \left(-1 - t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (+ 0.5 (* b 0.16666666666666666)))))
   (if (<= b -1.25)
     1.0
     (if (<= b 5e+102)
       (/
        (+ 2.0 (* b (+ -1.0 (* b (+ -0.5 (* b -0.16666666666666666))))))
        (+ 4.0 (* (* b (+ 1.0 t_0)) (* b (- -1.0 t_0)))))
       (/ -4.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double t_0 = b * (0.5 + (b * 0.16666666666666666));
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else if (b <= 5e+102) {
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (0.5d0 + (b * 0.16666666666666666d0))
    if (b <= (-1.25d0)) then
        tmp = 1.0d0
    else if (b <= 5d+102) then
        tmp = (2.0d0 + (b * ((-1.0d0) + (b * ((-0.5d0) + (b * (-0.16666666666666666d0))))))) / (4.0d0 + ((b * (1.0d0 + t_0)) * (b * ((-1.0d0) - t_0))))
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (0.5 + (b * 0.16666666666666666));
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0;
	} else if (b <= 5e+102) {
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (0.5 + (b * 0.16666666666666666))
	tmp = 0
	if b <= -1.25:
		tmp = 1.0
	elif b <= 5e+102:
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))))
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))
	tmp = 0.0
	if (b <= -1.25)
		tmp = 1.0;
	elseif (b <= 5e+102)
		tmp = Float64(Float64(2.0 + Float64(b * Float64(-1.0 + Float64(b * Float64(-0.5 + Float64(b * -0.16666666666666666)))))) / Float64(4.0 + Float64(Float64(b * Float64(1.0 + t_0)) * Float64(b * Float64(-1.0 - t_0)))));
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (0.5 + (b * 0.16666666666666666));
	tmp = 0.0;
	if (b <= -1.25)
		tmp = 1.0;
	elseif (b <= 5e+102)
		tmp = (2.0 + (b * (-1.0 + (b * (-0.5 + (b * -0.16666666666666666)))))) / (4.0 + ((b * (1.0 + t_0)) * (b * (-1.0 - t_0))));
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.25], 1.0, If[LessEqual[b, 5e+102], N[(N[(2.0 + N[(b * N[(-1.0 + N[(b * N[(-0.5 + N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(N[(b * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -1.25 < b < 5e102

   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 67.2%

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

   5. Simplified 67.2%

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

   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 57.0%

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

   8. Simplified 57.0%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\frac{2 \cdot 2 - \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)}{\color{blue}{2 - b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)}}} \]

      2. clear-num N/A

         \[\leadsto \frac{2 - b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)}{\color{blue}{2 \cdot 2 - \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)}} \]

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

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

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 - \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \cdot \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)}} \]

   if 5e102 < b

   1. Initial program 97.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 68.6%

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

   8. Simplified 68.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

       10. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 68.6%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   14. Simplified 100.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 + b \cdot \left(-1 + b \cdot \left(-0.5 + b \cdot -0.16666666666666666\right)\right)}{4 + \left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \cdot \left(b \cdot \left(-1 - b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 420:\\ \;\;\;\;0.5 + b \cdot \left(-0.25 + b \cdot \left(b \cdot 0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+80}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.4)
   1.0
   (if (<= b 420.0)
     (+ 0.5 (* b (+ -0.25 (* b (* b 0.020833333333333332)))))
     (if (<= b 1.08e+80)
       (* a (* (* a a) -0.020833333333333332))
       (/ -4.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.4) {
		tmp = 1.0;
	} else if (b <= 420.0) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else if (b <= 1.08e+80) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.4d0)) then
        tmp = 1.0d0
    else if (b <= 420.0d0) then
        tmp = 0.5d0 + (b * ((-0.25d0) + (b * (b * 0.020833333333333332d0))))
    else if (b <= 1.08d+80) then
        tmp = a * ((a * a) * (-0.020833333333333332d0))
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.4) {
		tmp = 1.0;
	} else if (b <= 420.0) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else if (b <= 1.08e+80) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.4:
		tmp = 1.0
	elif b <= 420.0:
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))))
	elif b <= 1.08e+80:
		tmp = a * ((a * a) * -0.020833333333333332)
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.4)
		tmp = 1.0;
	elseif (b <= 420.0)
		tmp = Float64(0.5 + Float64(b * Float64(-0.25 + Float64(b * Float64(b * 0.020833333333333332)))));
	elseif (b <= 1.08e+80)
		tmp = Float64(a * Float64(Float64(a * a) * -0.020833333333333332));
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.4)
		tmp = 1.0;
	elseif (b <= 420.0)
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	elseif (b <= 1.08e+80)
		tmp = a * ((a * a) * -0.020833333333333332);
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.4], 1.0, If[LessEqual[b, 420.0], N[(0.5 + N[(b * N[(-0.25 + N[(b * N[(b * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e+80], N[(a * N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.39999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -2.39999999999999991 < b < 420

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   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 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 63.1%

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

   8. Simplified 63.1%

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

   if 420 < b < 1.08e80

   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 35.5%

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

   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 2.7%

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left({a}^{2} \cdot a\right) \]

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 35.7%

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

   11. Simplified 35.7%

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

   if 1.08e80 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   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 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

       10. *-lowering-*.f64 65.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   14. Simplified 95.7%

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

Alternative 9: 75.4% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.061:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 350:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+80}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.061)
   1.0
   (if (<= b 350.0)
     (+ 0.5 (* a 0.25))
     (if (<= b 1.08e+80)
       (* a (* (* a a) -0.020833333333333332))
       (/ -4.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.061) {
		tmp = 1.0;
	} else if (b <= 350.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.08e+80) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.061d0)) then
        tmp = 1.0d0
    else if (b <= 350.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.08d+80) then
        tmp = a * ((a * a) * (-0.020833333333333332d0))
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.061) {
		tmp = 1.0;
	} else if (b <= 350.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.08e+80) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.061:
		tmp = 1.0
	elif b <= 350.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.08e+80:
		tmp = a * ((a * a) * -0.020833333333333332)
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.061)
		tmp = 1.0;
	elseif (b <= 350.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.08e+80)
		tmp = Float64(a * Float64(Float64(a * a) * -0.020833333333333332));
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.061)
		tmp = 1.0;
	elseif (b <= 350.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.08e+80)
		tmp = a * ((a * a) * -0.020833333333333332);
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.061], 1.0, If[LessEqual[b, 350.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e+80], N[(a * N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.060999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 98.0%

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

   if -0.060999999999999999 < b < 350

   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 97.5%

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

   6. Taylor expanded in a 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 63.4%

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

   8. Simplified 63.4%

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

   if 350 < b < 1.08e80

   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 35.5%

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

   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 2.7%

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left({a}^{2} \cdot a\right) \]

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 35.7%

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

   11. Simplified 35.7%

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

   if 1.08e80 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   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 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left(b \cdot \color{blue}{b}\right)\right) \]

       10. *-lowering-*.f64 65.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

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

       6. *-lowering-*.f64 95.7%

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   14. Simplified 95.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.061:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 350:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+80}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.075:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 410:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.075)
   1.0
   (if (<= b 410.0)
     (+ 0.5 (* a 0.25))
     (if (<= b 1.85e+151)
       (* a (* (* a a) -0.020833333333333332))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.075) {
		tmp = 1.0;
	} else if (b <= 410.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.85e+151) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} 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 <= (-0.075d0)) then
        tmp = 1.0d0
    else if (b <= 410.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.85d+151) then
        tmp = a * ((a * a) * (-0.020833333333333332d0))
    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 <= -0.075) {
		tmp = 1.0;
	} else if (b <= 410.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.85e+151) {
		tmp = a * ((a * a) * -0.020833333333333332);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.075:
		tmp = 1.0
	elif b <= 410.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.85e+151:
		tmp = a * ((a * a) * -0.020833333333333332)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.075)
		tmp = 1.0;
	elseif (b <= 410.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.85e+151)
		tmp = Float64(a * Float64(Float64(a * a) * -0.020833333333333332));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.075)
		tmp = 1.0;
	elseif (b <= 410.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.85e+151)
		tmp = a * ((a * a) * -0.020833333333333332);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.075], 1.0, If[LessEqual[b, 410.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e+151], N[(a * N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -0.0749999999999999972

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 98.0%

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

   if -0.0749999999999999972 < b < 410

   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 97.5%

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

   6. Taylor expanded in a 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 63.4%

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

   8. Simplified 63.4%

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

   if 410 < b < 1.8499999999999999e151

   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 33.2%

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

   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 2.7%

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left({a}^{2} \cdot a\right) \]

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 38.9%

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

   11. Simplified 38.9%

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

   if 1.8499999999999999e151 < b

   1. Initial program 96.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 94.0%

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

   8. Simplified 94.0%

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

   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 94.0%

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

   11. Simplified 94.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.075:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 410:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot -0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.55)
   1.0
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.55) {
		tmp = 1.0;
	} 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 <= (-1.55d0)) then
        tmp = 1.0d0
    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 <= -1.55) {
		tmp = 1.0;
	} 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 <= -1.55:
		tmp = 1.0
	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 <= -1.55)
		tmp = 1.0;
	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 <= -1.55)
		tmp = 1.0;
	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, -1.55], 1.0, N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -1.55000000000000004 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   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 65.4%

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

   8. Simplified 65.4%

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

Alternative 12: 68.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.061:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.75:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.061) 1.0 (if (<= b 1.75) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.061) {
		tmp = 1.0;
	} else if (b <= 1.75) {
		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 <= (-0.061d0)) then
        tmp = 1.0d0
    else if (b <= 1.75d0) 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 <= -0.061) {
		tmp = 1.0;
	} else if (b <= 1.75) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.061:
		tmp = 1.0
	elif b <= 1.75:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.061)
		tmp = 1.0;
	elseif (b <= 1.75)
		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 <= -0.061)
		tmp = 1.0;
	elseif (b <= 1.75)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.061], 1.0, If[LessEqual[b, 1.75], 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 < -0.060999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 98.0%

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

   if -0.060999999999999999 < b < 1.75

   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 97.5%

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

   6. Taylor expanded in a 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 63.4%

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

   8. Simplified 63.4%

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

   if 1.75 < b

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 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 49.6%

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

   8. Simplified 49.6%

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

   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 49.6%

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

   11. Simplified 49.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.061:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.75:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.7% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.95:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.95) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.95) {
		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 <= (-0.95d0)) 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 <= -0.95) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.95:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.95)
		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 <= -0.95)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.95], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -0.94999999999999996 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 46.4%

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

   8. Simplified 46.4%

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

Alternative 14: 54.9% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.075:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.075) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.075) {
		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 <= (-0.075d0)) 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 <= -0.075) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.075:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.075)
		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 <= -0.075)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.075], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0749999999999999972

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in a around inf

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

   11. Simplified 98.0%

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

   if -0.0749999999999999972 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.4%

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

   6. Taylor expanded in a 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 46.4%

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

   8. Simplified 46.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.075:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.0% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.8) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.8) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -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 <= (-1.8d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.8) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.8:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.8)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.8)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.8], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -1.80000000000000004 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 45.8%

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

   8. Simplified 45.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.7% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.94:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.94) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.94) {
		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 <= (-0.94d0)) 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 <= -0.94) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.94:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.94)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.94)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.94], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -0.93999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   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 -0.93999999999999995 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 45.3%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 39.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

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

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

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

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

   3. exp-lowering-exp.f64 78.6%

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

5. Simplified 78.6%

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

6. Taylor expanded in b around 0

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

8. Simplified 40.3%

   \[\leadsto \color{blue}{0.5} \]
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 2024192 
(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))))