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

Percentage Accurate: 88.9% → 98.8%

Time: 10.2s

Alternatives: 9

Speedup: 1.0×

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: 88.9% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 9 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z_m}}{z_m \cdot x_m}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 9e+30)
     (/ (/ 1.0 (* x_m (fma z_m z_m 1.0))) y_m)
     (/ (/ (/ 1.0 z_m) (* z_m x_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 9e+30) {
		tmp = (1.0 / (x_m * fma(z_m, z_m, 1.0))) / y_m;
	} else {
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 9e+30)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z_m, z_m, 1.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / Float64(z_m * x_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 9e+30], N[(N[(1.0 / N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.9999999999999999e30

   1. Initial program 95.4%

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

      1. associate-/l/ 95.4%

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

      2. metadata-eval 95.4%

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

      3. associate-/r* 95.4%

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

      4. metadata-eval 95.4%

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

      5. neg-mul-1 95.4%

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

      6. distribute-neg-frac 95.4%

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

      7. distribute-frac-neg 95.4%

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

      8. distribute-frac-neg 95.4%

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

      9. distribute-neg-frac 95.4%

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

      10. metadata-eval 95.4%

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

      11. neg-mul-1 95.4%

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

      12. associate-/r* 95.4%

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

      13. metadata-eval 95.4%

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

      14. associate-/r* 95.4%

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

      15. sqr-neg 95.4%

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

      16. +-commutative 95.4%

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

      17. sqr-neg 95.4%

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

      18. fma-def 95.4%

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

   3. Simplified 95.4%

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

   if 8.9999999999999999e30 < z

   1. Initial program 84.4%

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

      1. associate-/l/ 79.8%

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

      2. metadata-eval 79.8%

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

      3. associate-/r* 79.8%

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

      4. metadata-eval 79.8%

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

      5. neg-mul-1 79.8%

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

      6. distribute-neg-frac 79.8%

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

      7. distribute-frac-neg 79.8%

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

      8. distribute-frac-neg 79.8%

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

      9. distribute-neg-frac 79.8%

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

      10. metadata-eval 79.8%

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

      11. neg-mul-1 79.8%

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

      12. associate-/r* 79.8%

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

      13. metadata-eval 79.8%

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

      14. associate-/r* 79.9%

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

      15. sqr-neg 79.9%

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

      16. +-commutative 79.9%

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

      17. sqr-neg 79.9%

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

      18. fma-def 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in z around inf 79.9%

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

      1. associate-/r* 79.8%

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

      2. *-un-lft-identity 79.8%

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

      3. unpow2 79.8%

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

      4. times-frac 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. associate-/r* 87.5%

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

      2. un-div-inv 87.6%

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

   9. Applied egg-rr 87.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (/ (* (/ 1.0 (hypot 1.0 z_m)) (/ (/ 1.0 x_m) (hypot 1.0 z_m))) y_m))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (((1.0 / hypot(1.0, z_m)) * ((1.0 / x_m) / hypot(1.0, z_m))) / y_m));
}

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (((1.0 / Math.hypot(1.0, z_m)) * ((1.0 / x_m) / Math.hypot(1.0, z_m))) / y_m));
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

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

   1. associate-/l/ 91.7%

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

   2. metadata-eval 91.7%

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

   3. associate-/r* 91.7%

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

   4. metadata-eval 91.7%

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

   5. neg-mul-1 91.7%

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

   6. distribute-neg-frac 91.7%

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

   7. distribute-frac-neg 91.7%

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

   8. distribute-frac-neg 91.7%

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

   9. distribute-neg-frac 91.7%

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

   10. metadata-eval 91.7%

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

   11. neg-mul-1 91.7%

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

   12. associate-/r* 91.7%

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

   13. metadata-eval 91.7%

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

   14. associate-/r* 91.7%

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

   15. sqr-neg 91.7%

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

   16. +-commutative 91.7%

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

   17. sqr-neg 91.7%

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

   18. fma-def 91.7%

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

3. Simplified 91.7%

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

   1. associate-/r* 91.7%

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

   2. *-un-lft-identity 91.7%

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

   3. add-sqr-sqrt 91.7%

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

   4. times-frac 91.7%

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

   5. fma-udef 91.7%

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

   6. +-commutative 91.7%

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

   7. hypot-1-def 91.7%

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

   8. fma-udef 91.7%

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

   9. +-commutative 91.7%

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

   10. hypot-1-def 95.0%

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

6. Applied egg-rr 95.0%

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

7. Final simplification 95.0%

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

Alternative 3: 98.4% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (* (/ 1.0 (hypot 1.0 z_m)) (/ 1.0 (* y_m (* (hypot 1.0 z_m) x_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / hypot(1.0, z_m)) * (1.0 / (y_m * (hypot(1.0, z_m) * x_m)))));
}

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / Math.hypot(1.0, z_m)) * (1.0 / (y_m * (Math.hypot(1.0, z_m) * x_m)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$95$m * N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.8%

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

   1. associate-/l/ 91.7%

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

   2. metadata-eval 91.7%

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

   3. associate-/r* 91.7%

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

   4. metadata-eval 91.7%

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

   5. neg-mul-1 91.7%

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

   6. distribute-neg-frac 91.7%

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

   7. distribute-frac-neg 91.7%

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

   8. distribute-frac-neg 91.7%

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

   9. distribute-neg-frac 91.7%

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

   10. metadata-eval 91.7%

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

   11. neg-mul-1 91.7%

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

   12. associate-/r* 91.7%

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

   13. metadata-eval 91.7%

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

   14. associate-/r* 91.7%

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

   15. sqr-neg 91.7%

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

   16. +-commutative 91.7%

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

   17. sqr-neg 91.7%

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

   18. fma-def 91.7%

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

3. Simplified 91.7%

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

   1. associate-/r* 91.7%

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

   2. *-un-lft-identity 91.7%

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

   3. add-sqr-sqrt 91.7%

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

   4. times-frac 91.7%

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

   5. fma-udef 91.7%

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

   6. +-commutative 91.7%

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

   7. hypot-1-def 91.7%

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

   8. fma-udef 91.7%

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

   9. +-commutative 91.7%

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

   10. hypot-1-def 95.0%

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

6. Applied egg-rr 95.0%

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

   1. associate-/l* 97.3%

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

   2. div-inv 97.1%

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

   3. div-inv 96.7%

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

   4. clear-num 96.7%

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

   5. div-inv 96.7%

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

   6. clear-num 96.8%

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

   7. /-rgt-identity 96.8%

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

8. Applied egg-rr 96.8%

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

9. Final simplification 96.8%

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

Alternative 4: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z_m}}{z_m \cdot x_m}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+42)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (/ (/ (/ 1.0 z_m) (* z_m x_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+42) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    real(8) :: tmp
    if ((z_m * z_m) <= 2d+42) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else
        tmp = ((1.0d0 / z_m) / (z_m * x_m)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+42) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else {
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e+42:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	else:
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+42)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / Float64(z_m * x_m)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+42)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	else
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+42], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000009e42

   1. Initial program 99.7%

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

   if 2.00000000000000009e42 < (*.f64 z z)

   1. Initial program 84.0%

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

      1. associate-/l/ 81.5%

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

      2. metadata-eval 81.5%

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

      3. associate-/r* 81.5%

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

      4. metadata-eval 81.5%

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

      5. neg-mul-1 81.5%

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

      6. distribute-neg-frac 81.5%

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

      7. distribute-frac-neg 81.5%

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

      8. distribute-frac-neg 81.5%

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

      9. distribute-neg-frac 81.5%

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

      10. metadata-eval 81.5%

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

      11. neg-mul-1 81.5%

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

      12. associate-/r* 81.5%

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

      13. metadata-eval 81.5%

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

      14. associate-/r* 81.6%

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

      15. sqr-neg 81.6%

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

      16. +-commutative 81.6%

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

      17. sqr-neg 81.6%

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

      18. fma-def 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in z around inf 81.6%

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

      1. associate-/r* 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. unpow2 81.5%

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

      4. times-frac 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. associate-/r* 89.0%

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

      2. un-div-inv 89.0%

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

   9. Applied egg-rr 89.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 (* z_m (* z_m x_m))) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (z_m * (z_m * x_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.2%

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

      1. associate-/l/ 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-/r* 95.8%

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

      4. metadata-eval 95.8%

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

      5. neg-mul-1 95.8%

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

      6. distribute-neg-frac 95.8%

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

      7. distribute-frac-neg 95.8%

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

      8. distribute-frac-neg 95.8%

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

      9. distribute-neg-frac 95.8%

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

      10. metadata-eval 95.8%

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

      11. neg-mul-1 95.8%

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

      12. associate-/r* 95.8%

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

      13. metadata-eval 95.8%

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

      14. associate-/r* 95.8%

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

      15. sqr-neg 95.8%

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

      16. +-commutative 95.8%

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

      17. sqr-neg 95.8%

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

      18. fma-def 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around 0 75.8%

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

   if 1 < z

   1. Initial program 86.0%

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

      1. associate-/l/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. associate-/r* 80.5%

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

      4. metadata-eval 80.5%

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

      5. neg-mul-1 80.5%

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

      6. distribute-neg-frac 80.5%

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

      7. distribute-frac-neg 80.5%

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

      8. distribute-frac-neg 80.5%

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

      9. distribute-neg-frac 80.5%

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

      10. metadata-eval 80.5%

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

      11. neg-mul-1 80.5%

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

      12. associate-/r* 80.5%

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

      13. metadata-eval 80.5%

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

      14. associate-/r* 80.5%

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

      15. sqr-neg 80.5%

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

      16. +-commutative 80.5%

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

      17. sqr-neg 80.5%

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

      18. fma-def 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-/r* 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. unpow2 80.5%

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

      4. times-frac 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. associate-/r* 87.4%

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

      2. *-commutative 87.4%

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

      3. frac-times 86.2%

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

      4. metadata-eval 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 78.6%

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

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0) (/ (/ 1.0 x_m) y_m) (/ (/ (/ 1.0 z_m) (* z_m x_m)) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z_m) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.2%

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

      1. associate-/l/ 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-/r* 95.8%

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

      4. metadata-eval 95.8%

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

      5. neg-mul-1 95.8%

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

      6. distribute-neg-frac 95.8%

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

      7. distribute-frac-neg 95.8%

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

      8. distribute-frac-neg 95.8%

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

      9. distribute-neg-frac 95.8%

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

      10. metadata-eval 95.8%

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

      11. neg-mul-1 95.8%

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

      12. associate-/r* 95.8%

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

      13. metadata-eval 95.8%

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

      14. associate-/r* 95.8%

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

      15. sqr-neg 95.8%

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

      16. +-commutative 95.8%

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

      17. sqr-neg 95.8%

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

      18. fma-def 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around 0 75.8%

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

   if 1 < z

   1. Initial program 86.0%

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

      1. associate-/l/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. associate-/r* 80.5%

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

      4. metadata-eval 80.5%

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

      5. neg-mul-1 80.5%

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

      6. distribute-neg-frac 80.5%

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

      7. distribute-frac-neg 80.5%

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

      8. distribute-frac-neg 80.5%

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

      9. distribute-neg-frac 80.5%

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

      10. metadata-eval 80.5%

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

      11. neg-mul-1 80.5%

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

      12. associate-/r* 80.5%

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

      13. metadata-eval 80.5%

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

      14. associate-/r* 80.5%

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

      15. sqr-neg 80.5%

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

      16. +-commutative 80.5%

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

      17. sqr-neg 80.5%

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

      18. fma-def 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-/r* 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. unpow2 80.5%

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

      4. times-frac 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. associate-/r* 87.4%

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

      2. un-div-inv 87.4%

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

   9. Applied egg-rr 87.4%

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

4. Final simplification 78.9%

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

Alternative 7: 98.1% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 1.0) (/ (/ 1.0 x_m) y_m) (/ (/ (/ 1.0 (* z_m x_m)) z_m) y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / (z_m * x_m)) / z_m) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / (z_m * x_m)) / z_m) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.2%

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

      1. associate-/l/ 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-/r* 95.8%

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

      4. metadata-eval 95.8%

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

      5. neg-mul-1 95.8%

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

      6. distribute-neg-frac 95.8%

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

      7. distribute-frac-neg 95.8%

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

      8. distribute-frac-neg 95.8%

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

      9. distribute-neg-frac 95.8%

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

      10. metadata-eval 95.8%

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

      11. neg-mul-1 95.8%

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

      12. associate-/r* 95.8%

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

      13. metadata-eval 95.8%

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

      14. associate-/r* 95.8%

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

      15. sqr-neg 95.8%

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

      16. +-commutative 95.8%

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

      17. sqr-neg 95.8%

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

      18. fma-def 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around 0 75.8%

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

   if 1 < z

   1. Initial program 86.0%

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

      1. associate-/l/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. associate-/r* 80.5%

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

      4. metadata-eval 80.5%

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

      5. neg-mul-1 80.5%

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

      6. distribute-neg-frac 80.5%

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

      7. distribute-frac-neg 80.5%

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

      8. distribute-frac-neg 80.5%

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

      9. distribute-neg-frac 80.5%

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

      10. metadata-eval 80.5%

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

      11. neg-mul-1 80.5%

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

      12. associate-/r* 80.5%

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

      13. metadata-eval 80.5%

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

      14. associate-/r* 80.5%

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

      15. sqr-neg 80.5%

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

      16. +-commutative 80.5%

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

      17. sqr-neg 80.5%

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

      18. fma-def 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-/r* 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. unpow2 80.5%

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

      4. times-frac 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. associate-*l/ 87.4%

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

      2. *-un-lft-identity 87.4%

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

      3. associate-/r* 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 78.9%

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

Alternative 8: 73.7% accurate, 1.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (if (<= z_m 1.0) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* x_m (* z_m y_m)))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (z_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (x_m * (z_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.2%

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

      1. associate-/l/ 95.8%

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

      2. metadata-eval 95.8%

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

      3. associate-/r* 95.8%

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

      4. metadata-eval 95.8%

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

      5. neg-mul-1 95.8%

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

      6. distribute-neg-frac 95.8%

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

      7. distribute-frac-neg 95.8%

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

      8. distribute-frac-neg 95.8%

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

      9. distribute-neg-frac 95.8%

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

      10. metadata-eval 95.8%

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

      11. neg-mul-1 95.8%

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

      12. associate-/r* 95.8%

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

      13. metadata-eval 95.8%

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

      14. associate-/r* 95.8%

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

      15. sqr-neg 95.8%

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

      16. +-commutative 95.8%

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

      17. sqr-neg 95.8%

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

      18. fma-def 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around 0 75.8%

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

   if 1 < z

   1. Initial program 86.0%

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

      1. associate-/l/ 80.5%

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

      2. metadata-eval 80.5%

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

      3. associate-/r* 80.5%

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

      4. metadata-eval 80.5%

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

      5. neg-mul-1 80.5%

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

      6. distribute-neg-frac 80.5%

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

      7. distribute-frac-neg 80.5%

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

      8. distribute-frac-neg 80.5%

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

      9. distribute-neg-frac 80.5%

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

      10. metadata-eval 80.5%

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

      11. neg-mul-1 80.5%

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

      12. associate-/r* 80.5%

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

      13. metadata-eval 80.5%

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

      14. associate-/r* 80.5%

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

      15. sqr-neg 80.5%

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

      16. +-commutative 80.5%

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

      17. sqr-neg 80.5%

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

      18. fma-def 80.5%

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

   3. Simplified 80.5%

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

      1. associate-/r* 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. add-sqr-sqrt 80.5%

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

      4. times-frac 80.5%

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

      5. fma-udef 80.5%

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

      6. +-commutative 80.5%

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

      7. hypot-1-def 80.5%

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

      8. fma-udef 80.5%

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

      9. +-commutative 80.5%

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

      10. hypot-1-def 87.4%

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

   6. Applied egg-rr 87.4%

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

   7. Taylor expanded in z around inf 87.4%

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

   8. Taylor expanded in z around 0 50.3%

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

      1. *-commutative 50.3%

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

   10. Simplified 50.3%

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

4. Final simplification 69.1%

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

Alternative 9: 58.8% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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 92.8%

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

   1. associate-/r* 92.4%

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

   2. sqr-neg 92.4%

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

   3. +-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. fma-def 92.4%

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

3. Simplified 92.4%

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

5. Taylor expanded in z around 0 62.6%

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

6. Final simplification 62.6%

   \[\leadsto \frac{1}{x \cdot y} \]
7. 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 2024013 
(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)))))