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

Percentage Accurate: 78.2% → 99.4%

Time: 12.4s

Alternatives: 13

Speedup: 10.0×

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: 78.2% 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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -42000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -42000.0) t_0 (if (<= x 0.000145) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -42000.0) {
		tmp = t_0;
	} else if (x <= 0.000145) {
		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(-y) / x
    if (x <= (-42000.0d0)) then
        tmp = t_0
    else if (x <= 0.000145d0) 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(-y) / x;
	double tmp;
	if (x <= -42000.0) {
		tmp = t_0;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -42000.0:
		tmp = t_0
	elif x <= 0.000145:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -42000.0)
		tmp = t_0;
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -42000.0)
		tmp = t_0;
	elseif (x <= 0.000145)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -42000.0], t$95$0, If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -42000 or 1.45e-4 < x

   1. Initial program 73.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

Alternative 2: 87.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5 + \mathsf{fma}\left(y, 0.16666666666666666 + \left(\frac{0.3333333333333333}{x \cdot x} + \frac{-0.5}{x}\right), \frac{-0.5}{x}\right), 1\right), 1\right)}}{x}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), \frac{-1}{x}\right), \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (/
           1.0
           (fma
            y
            (fma
             y
             (+
              0.5
              (fma
               y
               (+
                0.16666666666666666
                (+ (/ 0.3333333333333333 (* x x)) (/ -0.5 x)))
               (/ -0.5 x)))
             1.0)
            1.0))
          x)))
   (if (<= x -3.6e+261)
     t_0
     (if (<= x -42000.0)
       (fma
        y
        (fma (/ y x) (fma y -0.16666666666666666 0.5) (/ -1.0 x))
        (/ 1.0 x))
       (if (<= x 0.000145) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 / fma(y, fma(y, (0.5 + fma(y, (0.16666666666666666 + ((0.3333333333333333 / (x * x)) + (-0.5 / x))), (-0.5 / x))), 1.0), 1.0)) / x;
	double tmp;
	if (x <= -3.6e+261) {
		tmp = t_0;
	} else if (x <= -42000.0) {
		tmp = fma(y, fma((y / x), fma(y, -0.16666666666666666, 0.5), (-1.0 / x)), (1.0 / x));
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / fma(y, fma(y, Float64(0.5 + fma(y, Float64(0.16666666666666666 + Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(-0.5 / x))), Float64(-0.5 / x))), 1.0), 1.0)) / x)
	tmp = 0.0
	if (x <= -3.6e+261)
		tmp = t_0;
	elseif (x <= -42000.0)
		tmp = fma(y, fma(Float64(y / x), fma(y, -0.16666666666666666, 0.5), Float64(-1.0 / x)), Float64(1.0 / x));
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(y * N[(y * N[(0.5 + N[(y * N[(0.16666666666666666 + N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -3.6e+261], t$95$0, If[LessEqual[x, -42000.0], N[(y * N[(N[(y / x), $MachinePrecision] * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000018e261 or 1.45e-4 < x

   1. Initial program 68.2%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 68.2

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{1 + y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) + 1}}}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(y, 1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right), 1\right)}}}{x} \]

   7. Simplified 80.6%

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5 + \mathsf{fma}\left(y, 0.16666666666666666 + \left(\frac{0.3333333333333333}{x \cdot x} + \frac{-0.5}{x}\right), \frac{-0.5}{x}\right), 1\right), 1\right)}}}{x} \]

   if -3.60000000000000018e261 < x < -42000

   1. Initial program 83.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

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

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

   8. Simplified 82.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), \frac{-1}{x}\right), \frac{1}{x}\right)} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

Alternative 3: 86.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), \frac{-1}{x}\right), \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          -1.0
          (* x (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) -1.0)))))
   (if (<= x -3.9e+261)
     t_0
     (if (<= x -42000.0)
       (fma
        y
        (fma (/ y x) (fma y -0.16666666666666666 0.5) (/ -1.0 x))
        (/ 1.0 x))
       (if (<= x 0.000145) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0));
	double tmp;
	if (x <= -3.9e+261) {
		tmp = t_0;
	} else if (x <= -42000.0) {
		tmp = fma(y, fma((y / x), fma(y, -0.16666666666666666, 0.5), (-1.0 / x)), (1.0 / x));
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0)))
	tmp = 0.0
	if (x <= -3.9e+261)
		tmp = t_0;
	elseif (x <= -42000.0)
		tmp = fma(y, fma(Float64(y / x), fma(y, -0.16666666666666666, 0.5), Float64(-1.0 / x)), Float64(1.0 / x));
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e+261], t$95$0, If[LessEqual[x, -42000.0], N[(y * N[(N[(y / x), $MachinePrecision] * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999994e261 or 1.45e-4 < x

   1. Initial program 68.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 62.5

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 62.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]
   9. Step-by-step derivation

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

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

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

   10. Applied egg-rr 57.2%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 80.2%

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

   if -3.89999999999999994e261 < x < -42000

   1. Initial program 83.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

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

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

   8. Simplified 82.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), \frac{-1}{x}\right), \frac{1}{x}\right)} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+261}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), \frac{-1}{x}\right), \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y - y \cdot y}{x}, \frac{1 - y}{x}\right)\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          -1.0
          (* x (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) -1.0)))))
   (if (<= x -7.6e+261)
     t_0
     (if (<= x -42000.0)
       (fma y (/ (- y (* y y)) x) (/ (- 1.0 y) x))
       (if (<= x 0.000145) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0));
	double tmp;
	if (x <= -7.6e+261) {
		tmp = t_0;
	} else if (x <= -42000.0) {
		tmp = fma(y, ((y - (y * y)) / x), ((1.0 - y) / x));
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0)))
	tmp = 0.0
	if (x <= -7.6e+261)
		tmp = t_0;
	elseif (x <= -42000.0)
		tmp = fma(y, Float64(Float64(y - Float64(y * y)) / x), Float64(Float64(1.0 - y) / x));
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e+261], t$95$0, If[LessEqual[x, -42000.0], N[(y * N[(N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.6000000000000003e261 or 1.45e-4 < x

   1. Initial program 68.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 62.5

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 62.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]
   9. Step-by-step derivation

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

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

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

   10. Applied egg-rr 57.2%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 80.2%

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

   if -7.6000000000000003e261 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. mul-1-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   10. Simplified 82.3%

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

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+261}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y - y \cdot y}{x}, \frac{1 - y}{x}\right)\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          -1.0
          (* x (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) -1.0)))))
   (if (<= x -1.1e+262)
     t_0
     (if (<= x -42000.0)
       (/ (fma y (fma y (- 1.0 y) -1.0) 1.0) x)
       (if (<= x 0.000145) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0));
	double tmp;
	if (x <= -1.1e+262) {
		tmp = t_0;
	} else if (x <= -42000.0) {
		tmp = fma(y, fma(y, (1.0 - y), -1.0), 1.0) / x;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(x * fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), -1.0)))
	tmp = 0.0
	if (x <= -1.1e+262)
		tmp = t_0;
	elseif (x <= -42000.0)
		tmp = Float64(fma(y, fma(y, Float64(1.0 - y), -1.0), 1.0) / x);
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 / N[(x * N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e+262], t$95$0, If[LessEqual[x, -42000.0], N[(N[(y * N[(y * N[(1.0 - y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.10000000000000005e262 or 1.45e-4 < x

   1. Initial program 68.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      3. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 62.5

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{x} \]

   8. Simplified 62.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]
   9. Step-by-step derivation

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

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

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

   10. Applied egg-rr 57.2%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 80.2%

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

   if -1.10000000000000005e262 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, -1\right), 1\right)}{x} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

      8. --lowering--.f64 82.3

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}}{x} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+262}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - y}{1 - y \cdot y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+261)
   (/ 1.0 (fma y x x))
   (if (<= x -42000.0)
     (/ (fma y (fma y (- 1.0 y) -1.0) 1.0) x)
     (if (<= x 0.000145) (/ 1.0 x) (/ (/ (- 1.0 y) (- 1.0 (* y y))) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+261) {
		tmp = 1.0 / fma(y, x, x);
	} else if (x <= -42000.0) {
		tmp = fma(y, fma(y, (1.0 - y), -1.0), 1.0) / x;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = ((1.0 - y) / (1.0 - (y * y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+261)
		tmp = Float64(1.0 / fma(y, x, x));
	elseif (x <= -42000.0)
		tmp = Float64(fma(y, fma(y, Float64(1.0 - y), -1.0), 1.0) / x);
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(Float64(1.0 - y) / Float64(1.0 - Float64(y * y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+261], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -42000.0], N[(N[(y * N[(y * N[(1.0 - y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / N[(1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.99999999999999957e261

   1. Initial program 45.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 45.6

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

   4. Applied egg-rr 45.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 83.0

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

   7. Simplified 83.0%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.0

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

   9. Applied egg-rr 83.0%

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

   if -5.99999999999999957e261 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, -1\right), 1\right)}{x} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

      8. --lowering--.f64 82.3

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}}{x} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if 1.45e-4 < x

   1. Initial program 72.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 72.6

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-lowering-*.f64 75.4

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

   9. Applied egg-rr 75.4%

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

Alternative 7: 85.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.00014:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+262)
   (/ 1.0 (fma y x x))
   (if (<= x -42000.0)
     (/ (fma y (fma y (- 1.0 y) -1.0) 1.0) x)
     (if (<= x 0.00014) (/ 1.0 x) (/ -1.0 (* x (- -1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+262) {
		tmp = 1.0 / fma(y, x, x);
	} else if (x <= -42000.0) {
		tmp = fma(y, fma(y, (1.0 - y), -1.0), 1.0) / x;
	} else if (x <= 0.00014) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (x * (-1.0 - y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+262)
		tmp = Float64(1.0 / fma(y, x, x));
	elseif (x <= -42000.0)
		tmp = Float64(fma(y, fma(y, Float64(1.0 - y), -1.0), 1.0) / x);
	elseif (x <= 0.00014)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 - y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e+262], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -42000.0], N[(N[(y * N[(y * N[(1.0 - y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.00014], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(x * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.04999999999999995e262

   1. Initial program 45.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 45.6

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

   4. Applied egg-rr 45.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 83.0

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

   7. Simplified 83.0%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.0

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

   9. Applied egg-rr 83.0%

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

   if -1.04999999999999995e262 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, -1\right), 1\right)}{x} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

      8. --lowering--.f64 82.3

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{1 - y}, -1\right), 1\right)}{x} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}}{x} \]

   if -42000 < x < 1.3999999999999999e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 72.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 72.6

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

      1. div-inv N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

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

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

      11. neg-lowering-neg.f64 75.2

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

   9. Applied egg-rr 75.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+262}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 1 - y, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.00014:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, 1 - y\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+261)
   (/ 1.0 (fma y x x))
   (if (<= x -42000.0)
     (/ (fma y y (- 1.0 y)) x)
     (if (<= x 0.000145) (/ 1.0 x) (/ -1.0 (* x (- -1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+261) {
		tmp = 1.0 / fma(y, x, x);
	} else if (x <= -42000.0) {
		tmp = fma(y, y, (1.0 - y)) / x;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (x * (-1.0 - y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+261)
		tmp = Float64(1.0 / fma(y, x, x));
	elseif (x <= -42000.0)
		tmp = Float64(fma(y, y, Float64(1.0 - y)) / x);
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 - y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e+261], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -42000.0], N[(N[(y * y + N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(x * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.7999999999999997e261

   1. Initial program 45.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 45.6

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

   4. Applied egg-rr 45.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 83.0

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

   7. Simplified 83.0%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 83.0

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

   9. Applied egg-rr 83.0%

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

   if -4.7999999999999997e261 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-+l+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{1 - y}\right)}{x} \]

      12. --lowering--.f64 79.1

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{1 - y}\right)}{x} \]

   10. Simplified 79.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, 1 - y\right)}}{x} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if 1.45e-4 < x

   1. Initial program 72.6%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 72.6

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 73.7

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

   7. Simplified 73.7%

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

      1. div-inv N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

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

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

      11. neg-lowering-neg.f64 75.2

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

   9. Applied egg-rr 75.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+261}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, 1 - y\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(-1 - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -42000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, 1 - y\right)}{x}\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma y x x))))
   (if (<= x -5e+261)
     t_0
     (if (<= x -42000.0)
       (/ (fma y y (- 1.0 y)) x)
       (if (<= x 0.000145) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(y, x, x);
	double tmp;
	if (x <= -5e+261) {
		tmp = t_0;
	} else if (x <= -42000.0) {
		tmp = fma(y, y, (1.0 - y)) / x;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(y, x, x))
	tmp = 0.0
	if (x <= -5e+261)
		tmp = t_0;
	elseif (x <= -42000.0)
		tmp = Float64(fma(y, y, Float64(1.0 - y)) / x);
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+261], t$95$0, If[LessEqual[x, -42000.0], N[(N[(y * y + N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000001e261 or 1.45e-4 < x

   1. Initial program 68.2%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 68.2

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 75.2

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

   7. Simplified 75.2%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 76.5

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

   9. Applied egg-rr 76.5%

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

   if -5.0000000000000001e261 < x < -42000

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 83.9

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

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 60.8

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. associate-+l+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{1 - y}\right)}{x} \]

      12. --lowering--.f64 79.1

          \[\leadsto \frac{\mathsf{fma}\left(y, y, \color{blue}{1 - y}\right)}{x} \]

   10. Simplified 79.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, y, 1 - y\right)}}{x} \]

   if -42000 < x < 1.45e-4

   1. Initial program 79.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

Alternative 10: 81.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.000145:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma y x x))))
   (if (<= x -7.2e+176) t_0 (if (<= x 0.000145) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(y, x, x);
	double tmp;
	if (x <= -7.2e+176) {
		tmp = t_0;
	} else if (x <= 0.000145) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(y, x, x))
	tmp = 0.0
	if (x <= -7.2e+176)
		tmp = t_0;
	elseif (x <= 0.000145)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+176], t$95$0, If[LessEqual[x, 0.000145], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999983e176 or 1.45e-4 < x

   1. Initial program 67.3%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 67.3

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

   4. Applied egg-rr 67.3%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 72.5

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

   7. Simplified 72.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 73.6

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

   9. Applied egg-rr 73.6%

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

   if -7.19999999999999983e176 < x < 1.45e-4

   1. Initial program 83.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 87.5%

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

Alternative 11: 78.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 135:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 135.0) (/ 1.0 x) (/ x (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 135.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 135.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 135.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 135.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 135.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 135.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 135.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 135

   1. Initial program 84.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 82.2%

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

   if 135 < y

   1. Initial program 41.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 2.7

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

   5. Simplified 2.7%

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

      1. frac-sub N/A

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

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

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

      3. *-lft-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-lowering-*.f64 12.0

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

   7. Applied egg-rr 12.0%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 63.1%

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

Alternative 12: 75.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e+92) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+92) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d+92) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+92) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e+92:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e+92)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e+92)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e+92], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.20000000000000025e92

   1. Initial program 81.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 78.6%

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

   if 3.20000000000000025e92 < y

   1. Initial program 42.1%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. clear-num N/A

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

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

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

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

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

      18. +-lowering-+.f64 39.5

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

   4. Applied egg-rr 39.5%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 53.8

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

   7. Simplified 53.8%

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

   8. Taylor expanded in y around inf

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

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

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

      2. *-lowering-*.f64 57.5

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

   10. Simplified 57.5%

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

Alternative 13: 75.5% accurate, 19.3× 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 75.8%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 72.6%

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

Developer Target 1: 78.2% accurate, 0.7× 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 2024205 
(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))