Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.2% → 99.6%

Time: 12.4s

Alternatives: 13

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{{x_m}^{-0.5}}{z}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x_m}}}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{1}{y_m}\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (pow x_m -0.5) z)))
   (*
    y_s
    (*
     x_s
     (if (<= (* z z) 5e+189)
       (/ (/ 1.0 (/ (fma z z 1.0) (/ 1.0 x_m))) y_m)
       (* t_0 (* t_0 (/ 1.0 y_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = pow(x_m, -0.5) / z;
	double tmp;
	if ((z * z) <= 5e+189) {
		tmp = (1.0 / (fma(z, z, 1.0) / (1.0 / x_m))) / y_m;
	} else {
		tmp = t_0 * (t_0 * (1.0 / y_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64((x_m ^ -0.5) / z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+189)
		tmp = Float64(Float64(1.0 / Float64(fma(z, z, 1.0) / Float64(1.0 / x_m))) / y_m);
	else
		tmp = Float64(t_0 * Float64(t_0 * Float64(1.0 / y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Power[x$95$m, -0.5], $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+189], N[(N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e189

   1. Initial program 98.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 98.6%

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

      2. metadata-eval 98.6%

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

      3. associate-/r* 98.6%

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

      4. metadata-eval 98.6%

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

      5. neg-mul-1 98.6%

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

      6. distribute-neg-frac 98.6%

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

      7. distribute-frac-neg 98.6%

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

      8. distribute-frac-neg 98.6%

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

      9. distribute-neg-frac 98.6%

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

      10. metadata-eval 98.6%

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

      11. neg-mul-1 98.6%

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

      12. associate-/r* 98.6%

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

      13. metadata-eval 98.6%

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

      14. associate-/r* 98.6%

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

      15. sqr-neg 98.6%

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

      16. +-commutative 98.6%

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

      17. sqr-neg 98.6%

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

      18. fma-def 98.6%

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

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. /-rgt-identity 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-/l* 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 5.0000000000000004e189 < (*.f64 z z)

   1. Initial program 76.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-/r* 77.1%

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

      4. metadata-eval 77.1%

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

      5. neg-mul-1 77.1%

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

      6. distribute-neg-frac 77.1%

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

      7. distribute-frac-neg 77.1%

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

      8. distribute-frac-neg 77.1%

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

      9. distribute-neg-frac 77.1%

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

      10. metadata-eval 77.1%

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

      11. neg-mul-1 77.1%

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

      12. associate-/r* 77.1%

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

      13. metadata-eval 77.1%

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

      14. associate-/r* 77.1%

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

      15. sqr-neg 77.1%

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

      16. +-commutative 77.1%

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

      17. sqr-neg 77.1%

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

      18. fma-def 77.1%

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

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 77.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. div-inv 77.1%

         \[\leadsto \color{blue}{\frac{1}{x \cdot {z}^{2}} \cdot \frac{1}{y}} \]

      2. add-sqr-sqrt 72.1%

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

      3. associate-*l* 72.1%

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

      4. associate-/r* 72.1%

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

      5. sqrt-div 42.8%

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

      6. inv-pow 42.8%

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

      7. sqrt-pow1 42.8%

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

      8. metadata-eval 42.8%

         \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{{z}^{2}}} \cdot \left(\sqrt{\frac{1}{x \cdot {z}^{2}}} \cdot \frac{1}{y}\right) \]

      9. unpow2 42.8%

         \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{z \cdot z}}} \cdot \left(\sqrt{\frac{1}{x \cdot {z}^{2}}} \cdot \frac{1}{y}\right) \]

      10. sqrt-prod 20.1%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \cdot \left(\sqrt{\frac{1}{x \cdot {z}^{2}}} \cdot \frac{1}{y}\right) \]

      11. add-sqr-sqrt 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{z}} \cdot \left(\sqrt{\frac{1}{x \cdot {z}^{2}}} \cdot \frac{1}{y}\right) \]

      12. associate-/r* 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \left(\sqrt{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}} \cdot \frac{1}{y}\right) \]

      13. sqrt-div 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \left(\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{{z}^{2}}}} \cdot \frac{1}{y}\right) \]

      14. inv-pow 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \left(\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{{z}^{2}}} \cdot \frac{1}{y}\right) \]

      15. sqrt-pow1 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{{z}^{2}}} \cdot \frac{1}{y}\right) \]

      16. metadata-eval 37.9%

          \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \left(\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{{z}^{2}}} \cdot \frac{1}{y}\right) \]

      17. unpow2 37.9%

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

      18. sqrt-prod 22.4%

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

      19. add-sqr-sqrt 53.6%

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

   7. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{z} \cdot \left(\frac{{x}^{-0.5}}{z} \cdot \frac{1}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{z} \cdot \left(\frac{{x}^{-0.5}}{z} \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x_m}}{\mathsf{hypot}\left(1, z\right)}}{y_m}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (* x_s (/ (* (/ 1.0 (hypot 1.0 z)) (/ (/ 1.0 x_m) (hypot 1.0 z))) y_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((1.0 / hypot(1.0, z)) * ((1.0 / x_m) / hypot(1.0, z))) / y_m));
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((1.0 / Math.hypot(1.0, z)) * ((1.0 / x_m) / Math.hypot(1.0, z))) / y_m));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (((1.0 / math.hypot(1.0, z)) * ((1.0 / x_m) / math.hypot(1.0, z))) / y_m))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(1.0 / hypot(1.0, z)) * Float64(Float64(1.0 / x_m) / hypot(1.0, z))) / y_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (((1.0 / hypot(1.0, z)) * ((1.0 / x_m) / hypot(1.0, z))) / y_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 91.9%

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

   2. metadata-eval 91.9%

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

   3. associate-/r* 91.9%

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

   4. metadata-eval 91.9%

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

   5. neg-mul-1 91.9%

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

   6. distribute-neg-frac 91.9%

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

   7. distribute-frac-neg 91.9%

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

   8. distribute-frac-neg 91.9%

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

   9. distribute-neg-frac 91.9%

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

   10. metadata-eval 91.9%

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

   11. neg-mul-1 91.9%

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

   12. associate-/r* 91.9%

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

   13. metadata-eval 91.9%

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

   14. associate-/r* 91.9%

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

   15. sqr-neg 91.9%

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

   16. +-commutative 91.9%

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

   17. sqr-neg 91.9%

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

   18. fma-def 91.9%

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

3. Simplified 91.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 91.9%

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

   2. *-un-lft-identity 91.9%

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

   3. add-sqr-sqrt 91.9%

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

   4. times-frac 91.9%

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

   5. fma-udef 91.9%

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

   6. +-commutative 91.9%

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

   7. hypot-1-def 91.9%

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

   8. fma-udef 91.9%

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

   9. +-commutative 91.9%

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

   10. hypot-1-def 95.3%

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

6. Applied egg-rr 95.3%

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

7. Final simplification 95.3%

   \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y} \]
8. Add Preprocessing

Alternative 3: 87.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y_m}}{x_m}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z \cdot x_m}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1000000000000.0)
     (/ (/ (/ 1.0 y_m) x_m) (fma z z 1.0))
     (/ (* (/ 1.0 (hypot 1.0 z)) (/ 1.0 (* z x_m))) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1000000000000.0) {
		tmp = ((1.0 / y_m) / x_m) / fma(z, z, 1.0);
	} else {
		tmp = ((1.0 / hypot(1.0, z)) * (1.0 / (z * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1000000000000.0)
		tmp = Float64(Float64(Float64(1.0 / y_m) / x_m) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / hypot(1.0, z)) * Float64(1.0 / Float64(z * x_m))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1000000000000.0], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e12

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 99.0%

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

      2. sqr-neg 99.0%

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

      3. +-commutative 99.0%

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

      4. sqr-neg 99.0%

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

      5. fma-def 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-/r* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

      6. associate-/l/ 99.7%

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

      7. associate-/l/ 99.0%

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

      8. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   if 1e12 < (*.f64 z z)

   1. Initial program 83.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 83.7%

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

      2. metadata-eval 83.7%

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

      3. associate-/r* 83.7%

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

      4. metadata-eval 83.7%

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

      5. neg-mul-1 83.7%

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

      6. distribute-neg-frac 83.7%

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

      7. distribute-frac-neg 83.7%

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

      8. distribute-frac-neg 83.7%

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

      9. distribute-neg-frac 83.7%

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

      10. metadata-eval 83.7%

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

      11. neg-mul-1 83.7%

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

      12. associate-/r* 83.7%

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

      13. metadata-eval 83.7%

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

      14. associate-/r* 83.7%

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

      15. sqr-neg 83.7%

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

      16. +-commutative 83.7%

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

      17. sqr-neg 83.7%

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

      18. fma-def 83.7%

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

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. add-sqr-sqrt 83.7%

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

      4. times-frac 83.7%

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

      5. fma-udef 83.7%

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

      6. +-commutative 83.7%

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

      7. hypot-1-def 83.7%

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

      8. fma-udef 83.7%

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

      9. +-commutative 83.7%

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

      10. hypot-1-def 90.7%

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

   6. Applied egg-rr 90.7%

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

   7. Taylor expanded in z around inf 68.6%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x_m}}}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x_m}}{\mathsf{hypot}\left(1, z\right) \cdot y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+238)
     (/ (/ 1.0 (/ (fma z z 1.0) (/ 1.0 x_m))) y_m)
     (/ (/ 1.0 (* z x_m)) (* (hypot 1.0 z) y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+238) {
		tmp = (1.0 / (fma(z, z, 1.0) / (1.0 / x_m))) / y_m;
	} else {
		tmp = (1.0 / (z * x_m)) / (hypot(1.0, z) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+238)
		tmp = Float64(Float64(1.0 / Float64(fma(z, z, 1.0) / Float64(1.0 / x_m))) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z * x_m)) / Float64(hypot(1.0, z) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+238], N[(N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999995e238

   1. Initial program 97.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 97.6%

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

      2. metadata-eval 97.6%

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

      3. associate-/r* 97.6%

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

      4. metadata-eval 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-neg-frac 97.6%

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

      7. distribute-frac-neg 97.6%

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

      8. distribute-frac-neg 97.6%

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

      9. distribute-neg-frac 97.6%

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

      10. metadata-eval 97.6%

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

      11. neg-mul-1 97.6%

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

      12. associate-/r* 97.6%

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

      13. metadata-eval 97.6%

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

      14. associate-/r* 97.6%

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

      15. sqr-neg 97.6%

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

      16. +-commutative 97.6%

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

      17. sqr-neg 97.6%

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

      18. fma-def 97.6%

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

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. /-rgt-identity 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-/l* 97.6%

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

   6. Applied egg-rr 97.6%

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

   if 4.99999999999999995e238 < (*.f64 z z)

   1. Initial program 77.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/r* 77.2%

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

      4. metadata-eval 77.2%

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

      5. neg-mul-1 77.2%

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

      6. distribute-neg-frac 77.2%

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

      7. distribute-frac-neg 77.2%

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

      8. distribute-frac-neg 77.2%

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

      9. distribute-neg-frac 77.2%

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

      10. metadata-eval 77.2%

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

      11. neg-mul-1 77.2%

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

      12. associate-/r* 77.2%

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

      13. metadata-eval 77.2%

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

      14. associate-/r* 77.2%

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

      15. sqr-neg 77.2%

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

      16. +-commutative 77.2%

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

      17. sqr-neg 77.2%

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

      18. fma-def 77.2%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. add-sqr-sqrt 77.2%

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

      4. times-frac 77.2%

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

      5. fma-udef 77.2%

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

      6. +-commutative 77.2%

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

      7. hypot-1-def 77.2%

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

      8. fma-udef 77.2%

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

      9. +-commutative 77.2%

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

      10. hypot-1-def 89.4%

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

   6. Applied egg-rr 89.4%

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

   7. Taylor expanded in z around inf 76.0%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 74.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}}{y}\right)\right)} \]

      2. expm1-udef 73.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}}{y}\right)} - 1} \]

      3. div-inv 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}\right) \cdot \frac{1}{y}}\right)} - 1 \]

      4. associate-*l/ 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{1}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{1}{y}\right)} - 1 \]

      5. *-un-lft-identity 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{x \cdot z}}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y}\right)} - 1 \]

      6. frac-times 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x \cdot z} \cdot 1}{\mathsf{hypot}\left(1, z\right) \cdot y}}\right)} - 1 \]

      7. associate-/r/ 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{\frac{x \cdot z}{1}}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

      8. clear-num 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{x \cdot z}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

      9. associate-/r* 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\frac{1}{x}}{z}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

   9. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{1}{x}}{z}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 78.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{x}}{z}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)\right)} \]

       2. expm1-log1p 79.5%

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

       3. associate-/r* 79.5%

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

       4. *-commutative 79.5%

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

   11. Simplified 79.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot z}}{y \cdot \mathsf{hypot}\left(1, z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{\mathsf{hypot}\left(1, z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x_m}}{\mathsf{hypot}\left(1, z\right) \cdot y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+238)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (/ (/ 1.0 (* z x_m)) (* (hypot 1.0 z) y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+238) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = (1.0 / (z * x_m)) / (hypot(1.0, z) * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+238)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z * x_m)) / Float64(hypot(1.0, z) * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+238], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999995e238

   1. Initial program 97.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 97.6%

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

      2. metadata-eval 97.6%

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

      3. associate-/r* 97.6%

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

      4. metadata-eval 97.6%

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

      5. neg-mul-1 97.6%

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

      6. distribute-neg-frac 97.6%

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

      7. distribute-frac-neg 97.6%

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

      8. distribute-frac-neg 97.6%

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

      9. distribute-neg-frac 97.6%

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

      10. metadata-eval 97.6%

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

      11. neg-mul-1 97.6%

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

      12. associate-/r* 97.6%

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

      13. metadata-eval 97.6%

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

      14. associate-/r* 97.6%

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

      15. sqr-neg 97.6%

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

      16. +-commutative 97.6%

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

      17. sqr-neg 97.6%

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

      18. fma-def 97.6%

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

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   if 4.99999999999999995e238 < (*.f64 z z)

   1. Initial program 77.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/r* 77.2%

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

      4. metadata-eval 77.2%

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

      5. neg-mul-1 77.2%

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

      6. distribute-neg-frac 77.2%

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

      7. distribute-frac-neg 77.2%

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

      8. distribute-frac-neg 77.2%

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

      9. distribute-neg-frac 77.2%

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

      10. metadata-eval 77.2%

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

      11. neg-mul-1 77.2%

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

      12. associate-/r* 77.2%

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

      13. metadata-eval 77.2%

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

      14. associate-/r* 77.2%

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

      15. sqr-neg 77.2%

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

      16. +-commutative 77.2%

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

      17. sqr-neg 77.2%

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

      18. fma-def 77.2%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. add-sqr-sqrt 77.2%

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

      4. times-frac 77.2%

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

      5. fma-udef 77.2%

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

      6. +-commutative 77.2%

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

      7. hypot-1-def 77.2%

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

      8. fma-udef 77.2%

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

      9. +-commutative 77.2%

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

      10. hypot-1-def 89.4%

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

   6. Applied egg-rr 89.4%

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

   7. Taylor expanded in z around inf 76.0%

      \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{1}{x \cdot z}}}{y} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 74.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}}{y}\right)\right)} \]

      2. expm1-udef 73.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}}{y}\right)} - 1} \]

      3. div-inv 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot z}\right) \cdot \frac{1}{y}}\right)} - 1 \]

      4. associate-*l/ 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{1}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{1}{y}\right)} - 1 \]

      5. *-un-lft-identity 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{x \cdot z}}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y}\right)} - 1 \]

      6. frac-times 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x \cdot z} \cdot 1}{\mathsf{hypot}\left(1, z\right) \cdot y}}\right)} - 1 \]

      7. associate-/r/ 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{\frac{x \cdot z}{1}}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

      8. clear-num 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{x \cdot z}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

      9. associate-/r* 73.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\frac{1}{x}}{z}}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1 \]

   9. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{1}{x}}{z}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 78.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{1}{x}}{z}}{\mathsf{hypot}\left(1, z\right) \cdot y}\right)\right)} \]

       2. expm1-log1p 79.5%

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

       3. associate-/r* 79.5%

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

       4. *-commutative 79.5%

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

   11. Simplified 79.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot z}}{y \cdot \mathsf{hypot}\left(1, z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{\mathsf{hypot}\left(1, z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y_m}}{x_m}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x_m}}{z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1000000000000.0)
     (/ (/ (/ 1.0 y_m) x_m) (fma z z 1.0))
     (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1000000000000.0) {
		tmp = ((1.0 / y_m) / x_m) / fma(z, z, 1.0);
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1000000000000.0)
		tmp = Float64(Float64(Float64(1.0 / y_m) / x_m) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1000000000000.0], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e12

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 99.0%

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

      2. sqr-neg 99.0%

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

      3. +-commutative 99.0%

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

      4. sqr-neg 99.0%

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

      5. fma-def 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-/r* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

      6. associate-/l/ 99.7%

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

      7. associate-/l/ 99.0%

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

      8. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   if 1e12 < (*.f64 z z)

   1. Initial program 83.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 83.7%

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

      2. metadata-eval 83.7%

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

      3. associate-/r* 83.7%

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

      4. metadata-eval 83.7%

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

      5. neg-mul-1 83.7%

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

      6. distribute-neg-frac 83.7%

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

      7. distribute-frac-neg 83.7%

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

      8. distribute-frac-neg 83.7%

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

      9. distribute-neg-frac 83.7%

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

      10. metadata-eval 83.7%

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

      11. neg-mul-1 83.7%

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

      12. associate-/r* 83.7%

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

      13. metadata-eval 83.7%

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

      14. associate-/r* 83.7%

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

      15. sqr-neg 83.7%

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

      16. +-commutative 83.7%

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

      17. sqr-neg 83.7%

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

      18. fma-def 83.7%

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

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 83.7%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 83.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 83.7%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 83.7%

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

      4. times-frac 90.7%

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

   7. Applied egg-rr 90.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1000000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x_m}}{z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+33)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+33) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+33) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+33) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e+33:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+33)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e+33)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+33], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e33

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   if 1.9999999999999999e33 < (*.f64 z z)

   1. Initial program 82.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 83.1%

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

      2. metadata-eval 83.1%

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

      3. associate-/r* 83.1%

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

      4. metadata-eval 83.1%

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

      5. neg-mul-1 83.1%

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

      6. distribute-neg-frac 83.1%

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

      7. distribute-frac-neg 83.1%

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

      8. distribute-frac-neg 83.1%

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

      9. distribute-neg-frac 83.1%

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

      10. metadata-eval 83.1%

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

      11. neg-mul-1 83.1%

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

      12. associate-/r* 83.1%

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

      13. metadata-eval 83.1%

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

      14. associate-/r* 83.0%

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

      15. sqr-neg 83.0%

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

      16. +-commutative 83.0%

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

      17. sqr-neg 83.0%

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

      18. fma-def 83.0%

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

   3. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 83.0%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 83.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 83.1%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 83.1%

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

      4. times-frac 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{z} \cdot \frac{\frac{1}{z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 1.0)
     (/ (/ 1.0 y_m) x_m)
     (* (/ (/ 1.0 x_m) z) (/ (/ 1.0 z) y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / x_m) / z) * ((1.0 / z) / y_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = ((1.0d0 / x_m) / z) * ((1.0d0 / z) / y_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / x_m) / z) * ((1.0 / z) / y_m);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = ((1.0 / x_m) / z) * ((1.0 / z) / y_m)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / z) * Float64(Float64(1.0 / z) / y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = ((1.0 / x_m) / z) * ((1.0 / z) / y_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 93.0%

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

      2. sqr-neg 93.0%

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

      3. +-commutative 93.0%

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

      4. sqr-neg 93.0%

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

      5. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-/r* 72.3%

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

   7. Simplified 72.3%

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

   if 1 < z

   1. Initial program 85.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 86.4%

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

      2. metadata-eval 86.4%

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

      3. associate-/r* 86.4%

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

      4. metadata-eval 86.4%

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

      5. neg-mul-1 86.4%

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

      6. distribute-neg-frac 86.4%

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

      7. distribute-frac-neg 86.4%

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

      8. distribute-frac-neg 86.4%

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

      9. distribute-neg-frac 86.4%

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

      10. metadata-eval 86.4%

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

      11. neg-mul-1 86.4%

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

      12. associate-/r* 86.4%

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

      13. metadata-eval 86.4%

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

      14. associate-/r* 86.4%

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

      15. sqr-neg 86.4%

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

      16. +-commutative 86.4%

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

      17. sqr-neg 86.4%

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

      18. fma-def 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 85.5%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. inv-pow 85.5%

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

      2. add-sqr-sqrt 42.2%

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

      3. unpow-prod-down 42.3%

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

      4. *-commutative 42.3%

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

      5. sqrt-prod 42.3%

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

      6. unpow2 42.3%

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

      7. sqrt-prod 42.3%

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

      8. add-sqr-sqrt 42.3%

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

      9. *-commutative 42.3%

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

      10. sqrt-prod 42.3%

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

      11. unpow2 42.3%

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

      12. sqrt-prod 45.6%

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

      13. add-sqr-sqrt 45.6%

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

   7. Applied egg-rr 45.6%

      \[\leadsto \frac{\color{blue}{{\left(z \cdot \sqrt{x}\right)}^{-1} \cdot {\left(z \cdot \sqrt{x}\right)}^{-1}}}{y} \]
   8. Step-by-step derivation

      1. pow-sqr 45.5%

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

      2. metadata-eval 45.5%

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

   9. Simplified 45.5%

      \[\leadsto \frac{\color{blue}{{\left(z \cdot \sqrt{x}\right)}^{-2}}}{y} \]
   10. Step-by-step derivation

       1. *-commutative 45.5%

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

       2. unpow-prod-down 42.3%

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

       3. sqrt-pow2 85.4%

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

       4. metadata-eval 85.4%

          \[\leadsto \frac{{x}^{\color{blue}{-1}} \cdot {z}^{-2}}{y} \]

       5. inv-pow 85.4%

          \[\leadsto \frac{\color{blue}{\frac{1}{x}} \cdot {z}^{-2}}{y} \]

   11. Applied egg-rr 85.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{x} \cdot {z}^{-2}}}{y} \]
   12. Step-by-step derivation

       1. associate-*l/ 85.5%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot {z}^{-2}}{x}}}{y} \]

       2. *-lft-identity 85.5%

          \[\leadsto \frac{\frac{\color{blue}{{z}^{-2}}}{x}}{y} \]

   13. Simplified 85.5%

       \[\leadsto \frac{\color{blue}{\frac{{z}^{-2}}{x}}}{y} \]
   14. Step-by-step derivation

       1. associate-/l/ 88.7%

          \[\leadsto \color{blue}{\frac{{z}^{-2}}{y \cdot x}} \]

       2. add-sqr-sqrt 88.6%

          \[\leadsto \frac{\color{blue}{\sqrt{{z}^{-2}} \cdot \sqrt{{z}^{-2}}}}{y \cdot x} \]

       3. times-frac 86.9%

          \[\leadsto \color{blue}{\frac{\sqrt{{z}^{-2}}}{y} \cdot \frac{\sqrt{{z}^{-2}}}{x}} \]

       4. sqrt-pow1 86.9%

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

       5. metadata-eval 86.9%

          \[\leadsto \frac{{z}^{\color{blue}{-1}}}{y} \cdot \frac{\sqrt{{z}^{-2}}}{x} \]

       6. unpow-1 86.9%

          \[\leadsto \frac{\color{blue}{\frac{1}{z}}}{y} \cdot \frac{\sqrt{{z}^{-2}}}{x} \]

       7. sqrt-pow1 96.8%

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

       8. metadata-eval 96.8%

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

       9. unpow-1 96.8%

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

       10. associate-/r* 96.1%

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

       11. associate-/l/ 96.8%

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

   15. Applied egg-rr 96.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \frac{\frac{1}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x_m}}{z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 1.0)
     (/ (/ 1.0 y_m) x_m)
     (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 93.0%

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

      2. sqr-neg 93.0%

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

      3. +-commutative 93.0%

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

      4. sqr-neg 93.0%

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

      5. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-/r* 72.3%

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

   7. Simplified 72.3%

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

   if 1 < z

   1. Initial program 85.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 86.4%

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

      2. metadata-eval 86.4%

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

      3. associate-/r* 86.4%

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

      4. metadata-eval 86.4%

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

      5. neg-mul-1 86.4%

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

      6. distribute-neg-frac 86.4%

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

      7. distribute-frac-neg 86.4%

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

      8. distribute-frac-neg 86.4%

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

      9. distribute-neg-frac 86.4%

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

      10. metadata-eval 86.4%

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

      11. neg-mul-1 86.4%

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

      12. associate-/r* 86.4%

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

      13. metadata-eval 86.4%

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

      14. associate-/r* 86.4%

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

      15. sqr-neg 86.4%

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

      16. +-commutative 86.4%

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

      17. sqr-neg 86.4%

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

      18. fma-def 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 85.5%

      \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot {z}^{2}}}}{y} \]
   6. Step-by-step derivation

      1. associate-/r* 85.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

      2. *-un-lft-identity 85.4%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{x}}}{{z}^{2}}}{y} \]

      3. unpow2 85.4%

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

      4. times-frac 90.8%

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

   7. Applied egg-rr 90.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x_m \cdot \left(z \cdot y_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z 1.0) (/ (/ 1.0 y_m) x_m) (/ 1.0 (* x_m (* z y_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (x_m * (z * y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / y_m) / x_m
    else
        tmp = 1.0d0 / (x_m * (z * y_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (x_m * (z * y_m));
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y_m) / x_m
	else:
		tmp = 1.0 / (x_m * (z * y_m))
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / y_m) / x_m);
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(z * y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / y_m) / x_m;
	else
		tmp = 1.0 / (x_m * (z * y_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 93.0%

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

      2. sqr-neg 93.0%

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

      3. +-commutative 93.0%

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

      4. sqr-neg 93.0%

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

      5. fma-def 93.0%

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

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-/r* 72.3%

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

   7. Simplified 72.3%

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

   if 1 < z

   1. Initial program 85.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 86.4%

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

      2. metadata-eval 86.4%

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

      3. associate-/r* 86.4%

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

      4. metadata-eval 86.4%

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

      5. neg-mul-1 86.4%

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

      6. distribute-neg-frac 86.4%

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

      7. distribute-frac-neg 86.4%

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

      8. distribute-frac-neg 86.4%

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

      9. distribute-neg-frac 86.4%

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

      10. metadata-eval 86.4%

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

      11. neg-mul-1 86.4%

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

      12. associate-/r* 86.4%

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

      13. metadata-eval 86.4%

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

      14. associate-/r* 86.4%

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

      15. sqr-neg 86.4%

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

      16. +-commutative 86.4%

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

      17. sqr-neg 86.4%

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

      18. fma-def 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 86.4%

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

      2. *-un-lft-identity 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. times-frac 86.4%

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

      5. fma-udef 86.4%

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

      6. +-commutative 86.4%

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

      7. hypot-1-def 86.4%

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

      8. fma-udef 86.4%

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

      9. +-commutative 86.4%

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

      10. hypot-1-def 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in z around inf 91.0%

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

   8. Taylor expanded in z around 0 41.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \frac{1}{x_m \cdot y_m}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (1.0 / (x_m * y_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (x_m * y_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/r* 91.3%

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

   2. sqr-neg 91.3%

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

   3. +-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. fma-def 91.3%

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

3. Simplified 91.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 59.9%

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

6. Final simplification 59.9%

   \[\leadsto \frac{1}{x \cdot y} \]
7. Add Preprocessing

Alternative 12: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \frac{\frac{1}{x_m}}{y_m}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 x_m) y_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * ((1.0d0 / x_m) / y_m))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / x_m) / y_m))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / x_m) / y_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / x_m) / y_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 91.9%

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

   2. metadata-eval 91.9%

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

   3. associate-/r* 91.9%

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

   4. metadata-eval 91.9%

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

   5. neg-mul-1 91.9%

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

   6. distribute-neg-frac 91.9%

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

   7. distribute-frac-neg 91.9%

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

   8. distribute-frac-neg 91.9%

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

   9. distribute-neg-frac 91.9%

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

   10. metadata-eval 91.9%

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

   11. neg-mul-1 91.9%

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

   12. associate-/r* 91.9%

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

   13. metadata-eval 91.9%

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

   14. associate-/r* 91.9%

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

   15. sqr-neg 91.9%

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

   16. +-commutative 91.9%

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

   17. sqr-neg 91.9%

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

   18. fma-def 91.9%

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

3. Simplified 91.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 60.2%

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

6. Final simplification 60.2%

   \[\leadsto \frac{\frac{1}{x}}{y} \]
7. Add Preprocessing

Alternative 13: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \frac{\frac{1}{y_m}}{x_m}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 y_m) x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * ((1.0d0 / y_m) / x_m))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / y_m) / x_m))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / y_m) / x_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / y_m) / x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.6%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/r* 91.3%

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

   2. sqr-neg 91.3%

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

   3. +-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. fma-def 91.3%

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

3. Simplified 91.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 59.9%

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

   1. *-commutative 59.9%

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

   2. associate-/r* 60.2%

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

7. Simplified 60.2%

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

8. Final simplification 60.2%

   \[\leadsto \frac{\frac{1}{y}}{x} \]
9. Add Preprocessing

Developer target: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t_0\\ t_2 := \frac{\frac{1}{y}}{t_0 \cdot x}\\ \mathbf{if}\;t_1 < -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024010 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))