Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 77.0% → 99.5%

Time: 11.8s

Alternatives: 9

Speedup: 12.3×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\frac{1}{e^{y}}}{x}\\ \mathbf{elif}\;x \leq 0.12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05)
   (/ (/ 1.0 (exp y)) x)
   (if (<= x 0.12) (/ 1.0 x) (/ (exp (- 0.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05) {
		tmp = (1.0 / exp(y)) / x;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = exp((0.0 - y)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (1.0d0 / exp(y)) / x
    else if (x <= 0.12d0) then
        tmp = 1.0d0 / x
    else
        tmp = exp((0.0d0 - y)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05) {
		tmp = (1.0 / Math.exp(y)) / x;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = Math.exp((0.0 - y)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05:
		tmp = (1.0 / math.exp(y)) / x
	elif x <= 0.12:
		tmp = 1.0 / x
	else:
		tmp = math.exp((0.0 - y)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(1.0 / exp(y)) / x);
	elseif (x <= 0.12)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(exp(Float64(0.0 - y)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (1.0 / exp(y)) / x;
	elseif (x <= 0.12)
		tmp = 1.0 / x;
	else
		tmp = exp((0.0 - y)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05], N[(N[(1.0 / N[Exp[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.12], N[(1.0 / x), $MachinePrecision], N[(N[Exp[N[(0.0 - y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   if -1.05000000000000004 < x < 0.12

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.12 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(y\right)\right), x\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{\frac{1}{e^{y}}}{x}\\ \mathbf{elif}\;x \leq 0.12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{0 - y}}{x}\\ \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- 0.0 y)) x)))
   (if (<= x -0.52) t_0 (if (<= x 0.12) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp((0.0 - y)) / x;
	double tmp;
	if (x <= -0.52) {
		tmp = t_0;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - y)) / x
    if (x <= (-0.52d0)) then
        tmp = t_0
    else if (x <= 0.12d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((0.0 - y)) / x;
	double tmp;
	if (x <= -0.52) {
		tmp = t_0;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((0.0 - y)) / x
	tmp = 0
	if x <= -0.52:
		tmp = t_0
	elif x <= 0.12:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(0.0 - y)) / x)
	tmp = 0.0
	if (x <= -0.52)
		tmp = t_0;
	elseif (x <= 0.12)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((0.0 - y)) / x;
	tmp = 0.0;
	if (x <= -0.52)
		tmp = t_0;
	elseif (x <= 0.12)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -0.52], t$95$0, If[LessEqual[x, 0.12], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.52000000000000002 or 0.12 < x

   1. Initial program 75.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 75.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(y\right)\right), x\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   if -0.52000000000000002 < x < 0.12

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \mathbf{elif}\;x \leq 0.12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.58)
   (+ (/ 1.0 x) (/ (* y (+ -1.0 (* y (+ (* y -0.16666666666666666) 0.5)))) x))
   (if (<= x 0.095)
     (/ 1.0 x)
     (/
      (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.58) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	} else if (x <= 0.095) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.58d0)) then
        tmp = (1.0d0 / x) + ((y * ((-1.0d0) + (y * ((y * (-0.16666666666666666d0)) + 0.5d0)))) / x)
    else if (x <= 0.095d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.58) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	} else if (x <= 0.095) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.58:
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x)
	elif x <= 0.095:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.58)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.16666666666666666) + 0.5)))) / x));
	elseif (x <= 0.095)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.58)
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	elseif (x <= 0.095)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.58], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.095], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.57999999999999996

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.4%

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

   10. Simplified 77.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. *-lowering-*.f64 77.4%

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

   12. Applied egg-rr 77.4%

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. associate-*l/ N/A

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

       6. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

       11. +-commutative N/A

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

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

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

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

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

       14. /-lowering-/.f64 77.4%

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

   14. Applied egg-rr 77.4%

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

   if -0.57999999999999996 < x < 0.095000000000000001

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.095000000000000001 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 85.5%

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

   12. Simplified 85.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.42)
   (+ (/ 1.0 x) (/ (* y (+ -1.0 (* y (+ (* y -0.16666666666666666) 0.5)))) x))
   (if (<= x 0.1) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y 0.5))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.42) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	} else if (x <= 0.1) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.42d0)) then
        tmp = (1.0d0 / x) + ((y * ((-1.0d0) + (y * ((y * (-0.16666666666666666d0)) + 0.5d0)))) / x)
    else if (x <= 0.1d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * 0.5d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.42) {
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	} else if (x <= 0.1) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.42:
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x)
	elif x <= 0.1:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.42)
		tmp = Float64(Float64(1.0 / x) + Float64(Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.16666666666666666) + 0.5)))) / x));
	elseif (x <= 0.1)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.42)
		tmp = (1.0 / x) + ((y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5)))) / x);
	elseif (x <= 0.1)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.42], N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.1], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.419999999999999984

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.4%

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

   10. Simplified 77.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. *-lowering-*.f64 77.4%

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

   12. Applied egg-rr 77.4%

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. associate-*l/ N/A

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

       6. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

       11. +-commutative N/A

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

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

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

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

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

       14. /-lowering-/.f64 77.4%

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

   14. Applied egg-rr 77.4%

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

   if -0.419999999999999984 < x < 0.10000000000000001

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.10000000000000001 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

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

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

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

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 79.8%

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

   12. Simplified 79.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{1}{x} + \frac{y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.55)
   (/ (+ 1.0 (* y (+ -1.0 (* y (+ (* y -0.16666666666666666) 0.5))))) x)
   (if (<= x 0.07) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y 0.5))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.55) {
		tmp = (1.0 + (y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5))))) / x;
	} else if (x <= 0.07) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.55d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * ((y * (-0.16666666666666666d0)) + 0.5d0))))) / x
    else if (x <= 0.07d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * 0.5d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.55) {
		tmp = (1.0 + (y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5))))) / x;
	} else if (x <= 0.07) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.55:
		tmp = (1.0 + (y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5))))) / x
	elif x <= 0.07:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.55)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.16666666666666666) + 0.5))))) / x);
	elseif (x <= 0.07)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.55)
		tmp = (1.0 + (y * (-1.0 + (y * ((y * -0.16666666666666666) + 0.5))))) / x;
	elseif (x <= 0.07)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.55], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.07], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.55000000000000004

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

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

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.4%

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

   10. Simplified 77.4%

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

   if -0.55000000000000004 < x < 0.070000000000000007

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.070000000000000007 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

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

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

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

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 79.8%

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

   12. Simplified 79.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(y \cdot -0.16666666666666666 + 0.5\right)\right)}{x}\\ \mathbf{elif}\;x \leq 0.07:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 0.12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y \cdot \left(1 + y \cdot 0.5\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.75)
   (/ (+ 1.0 (* y (+ -1.0 (* y 0.5)))) x)
   (if (<= x 0.12) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (* y (+ 1.0 (* y 0.5))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.75d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * 0.5d0)))) / x
    else if (x <= 0.12d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + (y * (1.0d0 + (y * 0.5d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.12) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.75:
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x
	elif x <= 0.12:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * 0.5)))) / x);
	elseif (x <= 0.12)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * 0.5))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	elseif (x <= 0.12)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + (y * (1.0 + (y * 0.5))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.75], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.12], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * N[(1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.75

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 70.0%

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

   10. Simplified 70.0%

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

   if -0.75 < x < 0.12

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.12 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

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

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

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

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 79.8%

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

   12. Simplified 79.8%

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

Alternative 7: 83.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 0.115:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.8)
   (/ (+ 1.0 (* y (+ -1.0 (* y 0.5)))) x)
   (if (<= x 0.115) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.8) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.115) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + y)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.8d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * 0.5d0)))) / x
    else if (x <= 0.115d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / (1.0d0 + y)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.8) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 0.115) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + y)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.8:
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x
	elif x <= 0.115:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / (1.0 + y)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.8)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * 0.5)))) / x);
	elseif (x <= 0.115)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.8)
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	elseif (x <= 0.115)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / (1.0 + y)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.8], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.115], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.80000000000000004

   1. Initial program 73.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 70.0%

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

   10. Simplified 70.0%

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

   if -0.80000000000000004 < x < 0.115000000000000005

   1. Initial program 87.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 87.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 97.2%

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

   7. Simplified 97.2%

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

   if 0.115000000000000005 < x

   1. Initial program 77.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 77.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

       1. +-lowering-+.f64 71.9%

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

   12. Simplified 71.9%

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

Alternative 8: 80.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{1 + y}}{x}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.044:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 (+ 1.0 y)) x)))
   (if (<= x -1.7e+76) t_0 (if (<= x 0.044) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / (1.0 + y)) / x;
	double tmp;
	if (x <= -1.7e+76) {
		tmp = t_0;
	} else if (x <= 0.044) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + y)) / x
    if (x <= (-1.7d+76)) then
        tmp = t_0
    else if (x <= 0.044d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / (1.0 + y)) / x;
	double tmp;
	if (x <= -1.7e+76) {
		tmp = t_0;
	} else if (x <= 0.044) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / (1.0 + y)) / x
	tmp = 0
	if x <= -1.7e+76:
		tmp = t_0
	elif x <= 0.044:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + y)) / x)
	tmp = 0.0
	if (x <= -1.7e+76)
		tmp = t_0;
	elseif (x <= 0.044)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / (1.0 + y)) / x;
	tmp = 0.0;
	if (x <= -1.7e+76)
		tmp = t_0;
	elseif (x <= 0.044)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.7e+76], t$95$0, If[LessEqual[x, 0.044], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e76 or 0.043999999999999997 < x

   1. Initial program 72.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 72.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{0 - y}\right), \color{blue}{x}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{0}}{e^{y}}\right), x\right) \]

      3. 1-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{y}}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{y}\right)\right), x\right) \]

      5. exp-lowering-exp.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{y}}}{x}} \]

   10. Taylor expanded in y around 0

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

       1. +-lowering-+.f64 68.5%

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

   12. Simplified 68.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{1 + y}}}{x} \]

   if -1.6999999999999999e76 < x < 0.043999999999999997

   1. Initial program 89.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 89.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 93.0%

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

   7. Simplified 93.0%

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

Alternative 9: 74.5% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 80.4%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

   3. exp-to-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

   6. +-lowering-+.f64 80.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

3. Simplified 80.5%

   \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 74.3%

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

7. Simplified 74.3%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
8. Add Preprocessing

Developer Target 1: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))