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

Percentage Accurate: 77.8% → 98.4%

Time: 11.5s

Alternatives: 7

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: 77.8% 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: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{0 - y}}{x}\\ \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- 0.0 y)) x)))
   (if (<= x -1400000000.0) t_0 (if (<= x 1.3e-45) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp((0.0 - y)) / x;
	double tmp;
	if (x <= -1400000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-45) {
		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 <= (-1400000000.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-45) 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 <= -1400000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-45) {
		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 <= -1400000000.0:
		tmp = t_0
	elif x <= 1.3e-45:
		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 <= -1400000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-45)
		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 <= -1400000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-45)
		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, -1400000000.0], t$95$0, If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e9 or 1.29999999999999993e-45 < x

   1. Initial program 74.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 74.0%

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

   3. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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 -1.4e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{\frac{x + \left(x \cdot y\right) \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1400000000.0)
   (/
    (/ (+ x (* (* x y) (+ -1.0 (* y (+ 0.5 (* y -0.16666666666666666)))))) x)
    x)
   (if (<= x 1.3e-45) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = ((x + ((x * y) * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1400000000.0d0)) then
        tmp = ((x + ((x * y) * ((-1.0d0) + (y * (0.5d0 + (y * (-0.16666666666666666d0))))))) / x) / x
    else if (x <= 1.3d-45) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = ((x + ((x * y) * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1400000000.0:
		tmp = ((x + ((x * y) * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x) / x
	elif x <= 1.3e-45:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1400000000.0)
		tmp = Float64(Float64(Float64(x + Float64(Float64(x * y) * Float64(-1.0 + Float64(y * Float64(0.5 + Float64(y * -0.16666666666666666)))))) / x) / x);
	elseif (x <= 1.3e-45)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1400000000.0)
		tmp = ((x + ((x * y) * (-1.0 + (y * (0.5 + (y * -0.16666666666666666)))))) / x) / x;
	elseif (x <= 1.3e-45)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1400000000.0], N[(N[(N[(x + N[(N[(x * y), $MachinePrecision] * N[(-1.0 + N[(y * N[(0.5 + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e9

   1. Initial program 69.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 74.8%

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

   10. Simplified 74.8%

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

       1. div-inv N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt1-in N/A

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

       4. div-inv N/A

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

       5. frac-add N/A

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

       6. associate-/r* N/A

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

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

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

   12. Applied egg-rr 78.3%

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

   if -1.4e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

   if 1.29999999999999993e-45 < x

   1. Initial program 78.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

      1. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - y \cdot y}{1 + y}}{x} \]

      2. associate-/l/ N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 75.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{\frac{x + \left(x \cdot y\right) \cdot \left(-1 + y \cdot \left(0.5 + y \cdot -0.16666666666666666\right)\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot \left(0.5 - y \cdot 0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1400000000.0)
   (/ (+ 1.0 (* y (+ -1.0 (* y (- 0.5 (* y 0.16666666666666666)))))) x)
   (if (<= x 1.3e-45) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 - (y * 0.16666666666666666)))))) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1400000000.0d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * (0.5d0 - (y * 0.16666666666666666d0)))))) / x
    else if (x <= 1.3d-45) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 - (y * 0.16666666666666666)))))) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1400000000.0:
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 - (y * 0.16666666666666666)))))) / x
	elif x <= 1.3e-45:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1400000000.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * Float64(0.5 - Float64(y * 0.16666666666666666)))))) / x);
	elseif (x <= 1.3e-45)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1400000000.0)
		tmp = (1.0 + (y * (-1.0 + (y * (0.5 - (y * 0.16666666666666666)))))) / x;
	elseif (x <= 1.3e-45)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1400000000.0], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * N[(0.5 - N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e9

   1. Initial program 69.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 74.8%

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

   10. Simplified 74.8%

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

   if -1.4e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

   if 1.29999999999999993e-45 < x

   1. Initial program 78.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

      1. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - y \cdot y}{1 + y}}{x} \]

      2. associate-/l/ N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 75.8%

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

Alternative 4: 83.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1400000000.0)
   (/ (+ 1.0 (* y (* y (* y -0.16666666666666666)))) x)
   (if (<= x 1.3e-45) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = (1.0 + (y * (y * (y * -0.16666666666666666)))) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1400000000.0d0)) then
        tmp = (1.0d0 + (y * (y * (y * (-0.16666666666666666d0))))) / x
    else if (x <= 1.3d-45) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = (1.0 + (y * (y * (y * -0.16666666666666666)))) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1400000000.0:
		tmp = (1.0 + (y * (y * (y * -0.16666666666666666)))) / x
	elif x <= 1.3e-45:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1400000000.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(y * -0.16666666666666666)))) / x);
	elseif (x <= 1.3e-45)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1400000000.0)
		tmp = (1.0 + (y * (y * (y * -0.16666666666666666)))) / x;
	elseif (x <= 1.3e-45)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1400000000.0], N[(N[(1.0 + N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e9

   1. Initial program 69.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   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. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 74.8%

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

   10. Simplified 74.8%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 73.6%

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

   13. Simplified 73.6%

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

   if -1.4e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

   if 1.29999999999999993e-45 < x

   1. Initial program 78.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

      1. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - y \cdot y}{1 + y}}{x} \]

      2. associate-/l/ N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 75.8%

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

Alternative 5: 82.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1400000000.0)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 1.3e-45) (/ 1.0 x) (/ 1.0 (* x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1400000000.0d0)) then
        tmp = ((x - (x * y)) / x) / x
    else if (x <= 1.3d-45) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1400000000.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1400000000.0:
		tmp = ((x - (x * y)) / x) / x
	elif x <= 1.3e-45:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1400000000.0)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 1.3e-45)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1400000000.0)
		tmp = ((x - (x * y)) / x) / x;
	elseif (x <= 1.3e-45)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1400000000.0], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e9

   1. Initial program 69.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{x} + \left(\mathsf{neg}\left(\frac{y}{x}\right)\right) \]

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 59.5%

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

   7. Simplified 59.5%

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

      1. frac-sub N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.7%

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

   9. Applied egg-rr 67.7%

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

   if -1.4e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

   if 1.29999999999999993e-45 < x

   1. Initial program 78.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 61.1%

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

   7. Simplified 61.1%

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

      1. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - y \cdot y}{1 + y}}{x} \]

      2. associate-/l/ N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.2%

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

   9. Applied egg-rr 66.2%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 75.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1400000000:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(y + 1\right)}\\ \mathbf{if}\;x \leq -1700000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x (+ y 1.0)))))
   (if (<= x -1700000000.0) t_0 (if (<= x 1.3e-45) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (y + 1.0));
	double tmp;
	if (x <= -1700000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-45) {
		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 / (x * (y + 1.0d0))
    if (x <= (-1700000000.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-45) 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 / (x * (y + 1.0));
	double tmp;
	if (x <= -1700000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-45) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * (y + 1.0))
	tmp = 0
	if x <= -1700000000.0:
		tmp = t_0
	elif x <= 1.3e-45:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(y + 1.0)))
	tmp = 0.0
	if (x <= -1700000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-45)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * (y + 1.0));
	tmp = 0.0;
	if (x <= -1700000000.0)
		tmp = t_0;
	elseif (x <= 1.3e-45)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1700000000.0], t$95$0, If[LessEqual[x, 1.3e-45], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e9 or 1.29999999999999993e-45 < x

   1. Initial program 74.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 60.4%

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

   7. Simplified 60.4%

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

      1. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - y \cdot y}{1 + y}}{x} \]

      2. associate-/l/ N/A

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

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

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

      4. metadata-eval N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 66.8%

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

   9. Applied egg-rr 66.8%

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

   10. Taylor expanded in y around 0

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

   12. Simplified 71.3%

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

   if -1.7e9 < x < 1.29999999999999993e-45

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

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

Alternative 7: 75.2% 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 77.6%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
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 77.6%

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

3. Simplified 77.6%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 73.4%

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

7. Simplified 73.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
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 2024138 
(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))