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

Percentage Accurate: 88.4% → 99.6%

Time: 11.9s

Alternatives: 14

Speedup: 0.5×

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: 88.4% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \sqrt{y\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot t\_0}}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (* z (sqrt y_m))))
   (*
    y_s
    (*
     x_s
     (if (<= (* z z) 2e+293)
       (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
       (/ (/ 1.0 (* x_m t_0)) t_0))))))

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 = z * sqrt(y_m);
	double tmp;
	if ((z * z) <= 2e+293) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (x_m * t_0)) / t_0;
	}
	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(z * sqrt(y_m))
	tmp = 0.0
	if (Float64(z * z) <= 2e+293)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * t_0)) / t_0);
	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[(z * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+293], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999998e293

   1. Initial program 98.1%

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

      1. remove-double-neg 98.1%

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

      2. distribute-lft-neg-out 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 97.4%

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

      6. associate-/l/ 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-neg-in 98.3%

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

      10. remove-double-neg 98.3%

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

      11. sqr-neg 98.3%

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

      12. +-commutative 98.3%

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

      13. sqr-neg 98.3%

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

      14. fma-define 98.3%

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

      15. *-commutative 98.3%

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

   3. Simplified 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 97.6%

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

      3. *-commutative 97.6%

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

      4. fma-undefine 97.6%

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

      5. +-commutative 97.6%

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

      6. associate-/l/ 98.1%

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

      7. *-un-lft-identity 98.1%

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

      8. add-sqr-sqrt 46.9%

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

      9. times-frac 47.0%

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

      10. +-commutative 47.0%

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

      11. fma-undefine 47.0%

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

      12. *-commutative 47.0%

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

      13. sqrt-prod 47.0%

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

      14. fma-undefine 47.0%

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

      15. +-commutative 47.0%

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

      16. hypot-1-def 47.0%

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

      17. +-commutative 47.0%

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

   6. Applied egg-rr 47.5%

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

   7. Taylor expanded in y around 0 97.6%

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

      1. associate-/r* 98.1%

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

      2. associate-/r* 97.6%

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

      3. associate-/l/ 97.4%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 95.8%

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

      6. +-commutative 95.8%

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

      7. unpow2 95.8%

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

      8. fma-define 95.8%

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

   9. Simplified 95.8%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 75.1%

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

      1. remove-double-neg 75.1%

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

      2. distribute-lft-neg-out 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. associate-/r* 74.7%

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

      5. associate-/l/ 74.6%

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

      6. associate-/l/ 74.6%

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

      7. distribute-lft-neg-out 74.6%

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

      8. distribute-rgt-neg-in 74.6%

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

      9. distribute-lft-neg-in 74.6%

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

      10. remove-double-neg 74.6%

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

      11. sqr-neg 74.6%

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

      12. +-commutative 74.6%

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

      13. sqr-neg 74.6%

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

      14. fma-define 74.6%

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

      15. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in z around inf 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. associate-/r* 74.6%

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

      3. pow-flip 75.7%

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

      4. metadata-eval 75.7%

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

      5. *-commutative 75.7%

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

   7. Applied egg-rr 75.7%

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

      1. *-lft-identity 75.7%

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

      2. *-commutative 75.7%

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

   9. Simplified 75.7%

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

       1. div-inv 75.7%

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

       2. metadata-eval 75.7%

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

       3. pow-flip 74.6%

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

       4. associate-/r* 74.7%

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

       5. times-frac 75.1%

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

       6. *-commutative 75.1%

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

       7. add-sqr-sqrt 40.8%

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

       8. times-frac 40.8%

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

       9. *-commutative 40.8%

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

       10. sqrt-prod 40.8%

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

       11. sqrt-pow1 40.8%

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

       12. metadata-eval 40.8%

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

       13. pow1 40.8%

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

       14. *-commutative 40.8%

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

       15. sqrt-prod 40.8%

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

       16. sqrt-pow1 51.2%

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

       17. metadata-eval 51.2%

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

       18. pow1 51.2%

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

   11. Applied egg-rr 51.2%

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

       1. associate-*l/ 51.2%

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

       2. *-lft-identity 51.2%

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

       3. associate-/l/ 51.2%

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

   13. Simplified 51.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot \sqrt{y}\right)}}{z \cdot \sqrt{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y\_m}\\ y\_s \cdot \left(x\_s \cdot \frac{\frac{1}{t\_0}}{t\_0 \cdot x\_m}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (* (hypot 1.0 z) (sqrt y_m))))
   (* y_s (* x_s (/ (/ 1.0 t_0) (* t_0 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) {
	double t_0 = hypot(1.0, z) * sqrt(y_m);
	return y_s * (x_s * ((1.0 / t_0) / (t_0 * x_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) {
	double t_0 = Math.hypot(1.0, z) * Math.sqrt(y_m);
	return y_s * (x_s * ((1.0 / t_0) / (t_0 * 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):
	t_0 = math.hypot(1.0, z) * math.sqrt(y_m)
	return y_s * (x_s * ((1.0 / t_0) / (t_0 * 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)
	t_0 = Float64(hypot(1.0, z) * sqrt(y_m))
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / t_0) / Float64(t_0 * 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)
	t_0 = hypot(1.0, z) * sqrt(y_m);
	tmp = y_s * (x_s * ((1.0 / t_0) / (t_0 * 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_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(t$95$0 * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 91.5%

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

   1. remove-double-neg 91.5%

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

   2. distribute-lft-neg-out 91.5%

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

   3. distribute-rgt-neg-in 91.5%

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

   4. associate-/r* 91.0%

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

   5. associate-/l/ 90.8%

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

   6. associate-/l/ 91.4%

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

   7. distribute-lft-neg-out 91.4%

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

   8. distribute-rgt-neg-in 91.4%

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

   9. distribute-lft-neg-in 91.4%

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

   10. remove-double-neg 91.4%

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

   11. sqr-neg 91.4%

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

   12. +-commutative 91.4%

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

   13. sqr-neg 91.4%

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

   14. fma-define 91.5%

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

   15. *-commutative 91.5%

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

3. Simplified 91.5%

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

   1. *-commutative 91.5%

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

   2. associate-*r* 91.1%

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

   3. *-commutative 91.1%

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

   4. fma-undefine 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-/l/ 91.5%

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

   7. *-un-lft-identity 91.5%

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

   8. add-sqr-sqrt 45.2%

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

   9. times-frac 45.2%

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

   10. +-commutative 45.2%

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

   11. fma-undefine 45.2%

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

   12. *-commutative 45.2%

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

   13. sqrt-prod 45.2%

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

   14. fma-undefine 45.2%

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

   15. +-commutative 45.2%

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

   16. hypot-1-def 45.2%

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

   17. +-commutative 45.2%

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

6. Applied egg-rr 48.5%

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

   1. associate-/l/ 48.5%

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

   2. associate-*r/ 48.5%

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

   3. *-rgt-identity 48.5%

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

   4. *-commutative 48.5%

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

8. Simplified 48.5%

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

9. Final simplification 48.5%

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

Alternative 3: 98.7% accurate, 0.1× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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) (hypot 1.0 z)) (/ 1.0 (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 / x_m) / hypot(1.0, z)) * (1.0 / 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 / x_m) / Math.hypot(1.0, z)) * (1.0 / 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 / x_m) / math.hypot(1.0, z)) * (1.0 / 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(Float64(1.0 / x_m) / hypot(1.0, z)) * Float64(1.0 / 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 / x_m) / hypot(1.0, z)) * (1.0 / 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[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. remove-double-neg 91.5%

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

   2. distribute-lft-neg-out 91.5%

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

   3. distribute-rgt-neg-in 91.5%

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

   4. associate-/r* 91.0%

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

   5. associate-/l/ 90.8%

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

   6. associate-/l/ 91.4%

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

   7. distribute-lft-neg-out 91.4%

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

   8. distribute-rgt-neg-in 91.4%

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

   9. distribute-lft-neg-in 91.4%

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

   10. remove-double-neg 91.4%

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

   11. sqr-neg 91.4%

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

   12. +-commutative 91.4%

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

   13. sqr-neg 91.4%

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

   14. fma-define 91.5%

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

   15. *-commutative 91.5%

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

3. Simplified 91.5%

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

   1. *-commutative 91.5%

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

   2. associate-*r* 91.1%

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

   3. *-commutative 91.1%

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

   4. fma-undefine 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-/l/ 91.5%

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

   7. *-un-lft-identity 91.5%

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

   8. add-sqr-sqrt 45.2%

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

   9. times-frac 45.2%

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

   10. +-commutative 45.2%

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

   11. fma-undefine 45.2%

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

   12. *-commutative 45.2%

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

   13. sqrt-prod 45.2%

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

   14. fma-undefine 45.2%

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

   15. +-commutative 45.2%

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

   16. hypot-1-def 45.2%

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

   17. +-commutative 45.2%

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

6. Applied egg-rr 48.5%

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

   1. *-commutative 48.5%

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

   2. associate-/r* 47.8%

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

   3. associate-/r* 47.8%

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

   4. frac-times 46.3%

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

   5. add-sqr-sqrt 94.5%

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

8. Applied egg-rr 94.5%

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

Alternative 4: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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+277)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (* (/ (/ 1.0 x_m) (hypot 1.0 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 * z) <= 5e+277) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (((1.0 / x_m) / hypot(1.0, 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 (Float64(z * z) <= 5e+277)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / hypot(1.0, z)) * Float64(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+277], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999982e277

   1. Initial program 98.6%

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

      1. remove-double-neg 98.6%

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

      2. distribute-lft-neg-out 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 97.3%

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

      6. associate-/l/ 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-neg-in 98.3%

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

      10. remove-double-neg 98.3%

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

      11. sqr-neg 98.3%

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

      12. +-commutative 98.3%

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

      13. sqr-neg 98.3%

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

      14. fma-define 98.3%

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

      15. *-commutative 98.3%

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

   3. Simplified 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.1%

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

      3. *-commutative 98.1%

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

      4. fma-undefine 98.1%

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

      5. +-commutative 98.1%

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

      6. associate-/l/ 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. add-sqr-sqrt 47.7%

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

      9. times-frac 47.7%

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

      10. +-commutative 47.7%

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

      11. fma-undefine 47.7%

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

      12. *-commutative 47.7%

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

      13. sqrt-prod 47.7%

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

      14. fma-undefine 47.7%

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

      15. +-commutative 47.7%

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

      16. hypot-1-def 47.7%

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

      17. +-commutative 47.7%

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

   6. Applied egg-rr 48.3%

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

   7. Taylor expanded in y around 0 98.1%

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

      1. associate-/r* 98.6%

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

      2. associate-/r* 97.6%

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

      3. associate-/l/ 97.3%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 96.3%

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

      6. +-commutative 96.3%

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

      7. unpow2 96.3%

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

      8. fma-define 96.3%

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

   9. Simplified 96.3%

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

   if 4.99999999999999982e277 < (*.f64 z z)

   1. Initial program 74.9%

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

      1. remove-double-neg 74.9%

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

      2. distribute-lft-neg-out 74.9%

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

      3. distribute-rgt-neg-in 74.9%

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

      4. associate-/r* 75.6%

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

      5. associate-/l/ 75.6%

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

      6. associate-/l/ 75.6%

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

      7. distribute-lft-neg-out 75.6%

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

      8. distribute-rgt-neg-in 75.6%

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

      9. distribute-lft-neg-in 75.6%

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

      10. remove-double-neg 75.6%

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

      11. sqr-neg 75.6%

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

      12. +-commutative 75.6%

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

      13. sqr-neg 75.6%

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

      14. fma-define 75.6%

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

      15. *-commutative 75.6%

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

   3. Simplified 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*r* 74.9%

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

      3. *-commutative 74.9%

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

      4. fma-undefine 74.9%

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

      5. +-commutative 74.9%

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

      6. associate-/l/ 74.9%

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

      7. *-un-lft-identity 74.9%

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

      8. add-sqr-sqrt 39.2%

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

      9. times-frac 39.2%

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

      10. +-commutative 39.2%

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

      11. fma-undefine 39.2%

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

      12. *-commutative 39.2%

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

      13. sqrt-prod 39.2%

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

      14. fma-undefine 39.2%

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

      15. +-commutative 39.2%

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

      16. hypot-1-def 39.2%

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

      17. +-commutative 39.2%

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

   6. Applied egg-rr 49.2%

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

      1. *-commutative 49.2%

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

      2. associate-/r* 49.2%

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

      3. associate-/r* 49.3%

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

      4. frac-times 45.4%

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

      5. add-sqr-sqrt 90.7%

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

   8. Applied egg-rr 90.7%

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

   9. Taylor expanded in z around inf 81.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{z} \cdot \frac{\frac{1}{y\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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+296)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (* (/ (/ 1.0 x_m) z) (/ (/ 1.0 y_m) z))))))

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+296) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / x_m) / z) * ((1.0 / y_m) / z);
	}
	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+296)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / z) * Float64(Float64(1.0 / y_m) / z));
	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+296], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e296

   1. Initial program 98.1%

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

      1. remove-double-neg 98.1%

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

      2. distribute-lft-neg-out 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 97.4%

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

      6. associate-/l/ 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-neg-in 98.3%

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

      10. remove-double-neg 98.3%

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

      11. sqr-neg 98.3%

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

      12. +-commutative 98.3%

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

      13. sqr-neg 98.3%

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

      14. fma-define 98.3%

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

      15. *-commutative 98.3%

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

   3. Simplified 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 97.6%

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

      3. *-commutative 97.6%

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

      4. fma-undefine 97.6%

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

      5. +-commutative 97.6%

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

      6. associate-/l/ 98.1%

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

      7. *-un-lft-identity 98.1%

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

      8. add-sqr-sqrt 47.0%

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

      9. times-frac 47.0%

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

      10. +-commutative 47.0%

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

      11. fma-undefine 47.0%

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

      12. *-commutative 47.0%

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

      13. sqrt-prod 47.0%

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

      14. fma-undefine 47.0%

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

      15. +-commutative 47.0%

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

      16. hypot-1-def 47.0%

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

      17. +-commutative 47.0%

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

   6. Applied egg-rr 47.5%

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

   7. Taylor expanded in y around 0 97.6%

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

      1. associate-/r* 98.1%

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

      2. associate-/r* 97.6%

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

      3. associate-/l/ 97.4%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 95.4%

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

      6. +-commutative 95.4%

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

      7. unpow2 95.4%

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

      8. fma-define 95.4%

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

   9. Simplified 95.4%

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

   if 5.0000000000000001e296 < (*.f64 z z)

   1. Initial program 74.5%

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

      1. remove-double-neg 74.5%

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

      2. distribute-lft-neg-out 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. associate-/r* 74.0%

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

      5. associate-/l/ 74.0%

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

      6. associate-/l/ 74.0%

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

      7. distribute-lft-neg-out 74.0%

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

      8. distribute-rgt-neg-in 74.0%

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

      9. distribute-lft-neg-in 74.0%

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

      10. remove-double-neg 74.0%

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

      11. sqr-neg 74.0%

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

      12. +-commutative 74.0%

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

      13. sqr-neg 74.0%

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

      14. fma-define 74.0%

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

      15. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in z around inf 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. associate-/r* 74.0%

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

      3. pow-flip 75.1%

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

      4. metadata-eval 75.1%

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

      5. *-commutative 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. *-lft-identity 75.1%

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

      2. *-commutative 75.1%

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

   9. Simplified 75.1%

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

       1. div-inv 75.1%

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

       2. metadata-eval 75.1%

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

       3. pow-flip 74.0%

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

       4. associate-/r* 74.0%

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

       5. times-frac 74.5%

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

       6. *-un-lft-identity 74.5%

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

       7. associate-/l/ 74.0%

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

       8. div-inv 74.0%

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

       9. unpow2 74.0%

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

       10. times-frac 98.5%

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \frac{\frac{1}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{z} \cdot \frac{\frac{1}{y\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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+296)
     (/ 1.0 (* (fma z z 1.0) (* y_m x_m)))
     (* (/ (/ 1.0 x_m) z) (/ (/ 1.0 y_m) z))))))

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+296) {
		tmp = 1.0 / (fma(z, z, 1.0) * (y_m * x_m));
	} else {
		tmp = ((1.0 / x_m) / z) * ((1.0 / y_m) / z);
	}
	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+296)
		tmp = Float64(1.0 / Float64(fma(z, z, 1.0) * Float64(y_m * x_m)));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / z) * Float64(Float64(1.0 / y_m) / z));
	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+296], N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e296

   1. Initial program 98.1%

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

      1. remove-double-neg 98.1%

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

      2. distribute-lft-neg-out 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. associate-/r* 97.6%

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

      5. associate-/l/ 97.4%

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

      6. associate-/l/ 98.3%

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

      7. distribute-lft-neg-out 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

      9. distribute-lft-neg-in 98.3%

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

      10. remove-double-neg 98.3%

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

      11. sqr-neg 98.3%

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

      12. +-commutative 98.3%

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

      13. sqr-neg 98.3%

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

      14. fma-define 98.3%

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

      15. *-commutative 98.3%

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

   3. Simplified 98.3%

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

   if 5.0000000000000001e296 < (*.f64 z z)

   1. Initial program 74.5%

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

      1. remove-double-neg 74.5%

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

      2. distribute-lft-neg-out 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. associate-/r* 74.0%

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

      5. associate-/l/ 74.0%

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

      6. associate-/l/ 74.0%

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

      7. distribute-lft-neg-out 74.0%

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

      8. distribute-rgt-neg-in 74.0%

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

      9. distribute-lft-neg-in 74.0%

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

      10. remove-double-neg 74.0%

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

      11. sqr-neg 74.0%

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

      12. +-commutative 74.0%

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

      13. sqr-neg 74.0%

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

      14. fma-define 74.0%

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

      15. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in z around inf 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. associate-/r* 74.0%

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

      3. pow-flip 75.1%

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

      4. metadata-eval 75.1%

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

      5. *-commutative 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. *-lft-identity 75.1%

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

      2. *-commutative 75.1%

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

   9. Simplified 75.1%

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

       1. div-inv 75.1%

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

       2. metadata-eval 75.1%

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

       3. pow-flip 74.0%

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

       4. associate-/r* 74.0%

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

       5. times-frac 74.5%

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

       6. *-un-lft-identity 74.5%

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

       7. associate-/l/ 74.0%

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

       8. div-inv 74.0%

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

       9. unpow2 74.0%

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

       10. times-frac 98.5%

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z} \cdot \frac{\frac{1}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y\_m \cdot x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 4.2e+26)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (if (<= z 2.1e+178)
       (* (/ 1.0 z) (/ (/ 1.0 (* y_m x_m)) 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 <= 4.2e+26) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else if (z <= 2.1e+178) {
		tmp = (1.0 / z) * ((1.0 / (y_m * x_m)) / 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 <= 4.2d+26) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else if (z <= 2.1d+178) then
        tmp = (1.0d0 / z) * ((1.0d0 / (y_m * x_m)) / 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 <= 4.2e+26) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else if (z <= 2.1e+178) {
		tmp = (1.0 / z) * ((1.0 / (y_m * x_m)) / 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 <= 4.2e+26:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	elif z <= 2.1e+178:
		tmp = (1.0 / z) * ((1.0 / (y_m * x_m)) / 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 (z <= 4.2e+26)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	elseif (z <= 2.1e+178)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / Float64(y_m * x_m)) / z));
	else
		tmp = Float64(Float64(1.0 / z) * 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 <= 4.2e+26)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	elseif (z <= 2.1e+178)
		tmp = (1.0 / z) * ((1.0 / (y_m * x_m)) / 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[z, 4.2e+26], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+178], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.2000000000000002e26

   1. Initial program 93.2%

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

   if 4.2000000000000002e26 < z < 2.0999999999999999e178

   1. Initial program 91.5%

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

      1. remove-double-neg 91.5%

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

      2. distribute-lft-neg-out 91.5%

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

      3. distribute-rgt-neg-in 91.5%

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

      4. associate-/r* 95.6%

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

      5. associate-/l/ 95.6%

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

      6. associate-/l/ 95.6%

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

      7. distribute-lft-neg-out 95.6%

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

      8. distribute-rgt-neg-in 95.6%

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

      9. distribute-lft-neg-in 95.6%

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

      10. remove-double-neg 95.6%

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

      11. sqr-neg 95.6%

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

      12. +-commutative 95.6%

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

      13. sqr-neg 95.6%

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

      14. fma-define 95.6%

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

      15. *-commutative 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in z around inf 95.6%

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

      1. *-un-lft-identity 95.6%

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

      2. associate-/r* 95.6%

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

      3. pow-flip 95.8%

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

      4. metadata-eval 95.8%

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

      5. *-commutative 95.8%

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

   7. Applied egg-rr 95.8%

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

      1. *-lft-identity 95.8%

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

      2. *-commutative 95.8%

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

   9. Simplified 95.8%

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

       1. div-inv 95.8%

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

       2. metadata-eval 95.8%

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

       3. pow-flip 95.6%

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

       4. associate-/r* 95.6%

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

       5. times-frac 91.5%

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

       6. *-un-lft-identity 91.5%

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

       7. associate-/l/ 95.6%

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

       8. associate-/r* 95.6%

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

       9. *-un-lft-identity 95.6%

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

       10. unpow2 95.6%

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

       11. times-frac 99.9%

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

       12. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   if 2.0999999999999999e178 < z

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

      4. associate-/r* 82.2%

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

      5. associate-/l/ 82.2%

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

      6. associate-/l/ 82.2%

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

      7. distribute-lft-neg-out 82.2%

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

      8. distribute-rgt-neg-in 82.2%

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

      9. distribute-lft-neg-in 82.2%

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

      10. remove-double-neg 82.2%

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

      11. sqr-neg 82.2%

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

      12. +-commutative 82.2%

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

      13. sqr-neg 82.2%

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

      14. fma-define 82.2%

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

      15. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in z around inf 82.2%

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

      1. *-un-lft-identity 82.2%

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

      2. associate-/r* 82.2%

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

      3. pow-flip 82.2%

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

      4. metadata-eval 82.2%

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

      5. *-commutative 82.2%

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

   7. Applied egg-rr 82.2%

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

      1. *-lft-identity 82.2%

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

      2. *-commutative 82.2%

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

   9. Simplified 82.2%

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

       1. div-inv 82.2%

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

       2. metadata-eval 82.2%

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

       3. pow-flip 82.2%

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

       4. associate-/r* 82.2%

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

       5. times-frac 82.7%

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

       6. *-un-lft-identity 82.7%

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

       7. associate-/l/ 82.2%

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

       8. associate-/r* 82.2%

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

       9. *-un-lft-identity 82.2%

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

       10. unpow2 82.2%

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

       11. times-frac 85.0%

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

       12. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   12. Taylor expanded in y around 0 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+178}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y \cdot x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \left(y\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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)
     (if (<= z 1.7e+178)
       (/ (/ 1.0 z) (* z (* 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 if (z <= 1.7e+178) {
		tmp = (1.0 / z) / (z * (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 if (z <= 1.7d+178) then
        tmp = (1.0d0 / z) / (z * (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 if (z <= 1.7e+178) {
		tmp = (1.0 / z) / (z * (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
	elif z <= 1.7e+178:
		tmp = (1.0 / z) / (z * (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);
	elseif (z <= 1.7e+178)
		tmp = Float64(Float64(1.0 / z) / Float64(z * Float64(y_m * x_m)));
	else
		tmp = Float64(Float64(1.0 / z) * 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;
	elseif (z <= 1.7e+178)
		tmp = (1.0 / z) / (z * (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], If[LessEqual[z, 1.7e+178], N[(N[(1.0 / z), $MachinePrecision] / N[(z * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. remove-double-neg 92.9%

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

      2. distribute-lft-neg-out 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. associate-/r* 92.4%

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

      5. associate-/l/ 92.1%

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

      6. associate-/l/ 92.7%

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

      7. distribute-lft-neg-out 92.7%

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

      8. distribute-rgt-neg-in 92.7%

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

      9. distribute-lft-neg-in 92.7%

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

      10. remove-double-neg 92.7%

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

      11. sqr-neg 92.7%

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

      12. +-commutative 92.7%

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

      13. sqr-neg 92.7%

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

      14. fma-define 92.7%

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

      15. *-commutative 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   if 1 < z < 1.7000000000000001e178

   1. Initial program 93.6%

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

      1. remove-double-neg 93.6%

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

      2. distribute-lft-neg-out 93.6%

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

      3. distribute-rgt-neg-in 93.6%

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

      4. associate-/r* 93.8%

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

      5. associate-/l/ 93.8%

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

      6. associate-/l/ 95.9%

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

      7. distribute-lft-neg-out 95.9%

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

      8. distribute-rgt-neg-in 95.9%

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

      9. distribute-lft-neg-in 95.9%

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

      10. remove-double-neg 95.9%

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

      11. sqr-neg 95.9%

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

      12. +-commutative 95.9%

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

      13. sqr-neg 95.9%

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

      14. fma-define 95.9%

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

      15. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in z around inf 93.3%

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

      1. *-un-lft-identity 93.3%

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

      2. associate-/r* 93.3%

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

      3. pow-flip 93.6%

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

      4. metadata-eval 93.6%

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

      5. *-commutative 93.6%

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

   7. Applied egg-rr 93.6%

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

      1. *-lft-identity 93.6%

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

      2. *-commutative 93.6%

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

   9. Simplified 93.6%

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

       1. div-inv 91.6%

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

       2. metadata-eval 91.6%

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

       3. pow-flip 91.3%

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

       4. associate-/r* 91.3%

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

       5. times-frac 91.1%

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

       6. *-un-lft-identity 91.1%

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

       7. associate-/l/ 91.2%

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

       8. associate-/r* 91.2%

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

       9. *-un-lft-identity 91.2%

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

       10. unpow2 91.2%

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

       11. times-frac 94.3%

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

       12. *-commutative 94.3%

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

   11. Applied egg-rr 94.3%

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

       1. clear-num 94.2%

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

       2. un-div-inv 94.3%

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

       3. div-inv 94.2%

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

       4. clear-num 96.3%

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

       5. /-rgt-identity 96.3%

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

       6. *-commutative 96.3%

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

   13. Applied egg-rr 96.3%

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

   if 1.7000000000000001e178 < z

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

      4. associate-/r* 82.2%

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

      5. associate-/l/ 82.2%

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

      6. associate-/l/ 82.2%

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

      7. distribute-lft-neg-out 82.2%

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

      8. distribute-rgt-neg-in 82.2%

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

      9. distribute-lft-neg-in 82.2%

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

      10. remove-double-neg 82.2%

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

      11. sqr-neg 82.2%

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

      12. +-commutative 82.2%

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

      13. sqr-neg 82.2%

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

      14. fma-define 82.2%

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

      15. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in z around inf 82.2%

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

      1. *-un-lft-identity 82.2%

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

      2. associate-/r* 82.2%

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

      3. pow-flip 82.2%

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

      4. metadata-eval 82.2%

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

      5. *-commutative 82.2%

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

   7. Applied egg-rr 82.2%

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

      1. *-lft-identity 82.2%

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

      2. *-commutative 82.2%

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

   9. Simplified 82.2%

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

       1. div-inv 82.2%

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

       2. metadata-eval 82.2%

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

       3. pow-flip 82.2%

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

       4. associate-/r* 82.2%

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

       5. times-frac 82.7%

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

       6. *-un-lft-identity 82.7%

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

       7. associate-/l/ 82.2%

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

       8. associate-/r* 82.2%

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

       9. *-un-lft-identity 82.2%

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

       10. unpow2 82.2%

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

       11. times-frac 85.0%

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

       12. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   12. Taylor expanded in y around 0 99.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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) (* z (* 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) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (1.0 / z) / (z * (y_m * x_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) / (z * (y_m * x_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) / (z * (y_m * x_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) / (z * (y_m * x_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(1.0 / z) / Float64(z * Float64(y_m * x_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) / (z * (y_m * x_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[(1.0 / z), $MachinePrecision] / N[(z * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. remove-double-neg 92.9%

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

      2. distribute-lft-neg-out 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. associate-/r* 92.4%

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

      5. associate-/l/ 92.1%

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

      6. associate-/l/ 92.7%

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

      7. distribute-lft-neg-out 92.7%

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

      8. distribute-rgt-neg-in 92.7%

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

      9. distribute-lft-neg-in 92.7%

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

      10. remove-double-neg 92.7%

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

      11. sqr-neg 92.7%

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

      12. +-commutative 92.7%

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

      13. sqr-neg 92.7%

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

      14. fma-define 92.7%

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

      15. *-commutative 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   if 1 < z

   1. Initial program 87.5%

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

      1. remove-double-neg 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.3%

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

      5. associate-/l/ 87.3%

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

      6. associate-/l/ 88.2%

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

      7. distribute-lft-neg-out 88.2%

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

      8. distribute-rgt-neg-in 88.2%

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

      9. distribute-lft-neg-in 88.2%

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

      10. remove-double-neg 88.2%

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

      11. sqr-neg 88.2%

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

      12. +-commutative 88.2%

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

      13. sqr-neg 88.2%

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

      14. fma-define 88.2%

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

      15. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in z around inf 87.1%

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

      1. *-un-lft-identity 87.1%

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

      2. associate-/r* 87.1%

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

      3. pow-flip 87.2%

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

      4. metadata-eval 87.2%

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

      5. *-commutative 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. *-lft-identity 87.2%

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

      2. *-commutative 87.2%

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

   9. Simplified 87.2%

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

       1. div-inv 86.3%

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

       2. metadata-eval 86.3%

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

       3. pow-flip 86.2%

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

       4. associate-/r* 86.2%

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

       5. times-frac 86.4%

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

       6. *-un-lft-identity 86.4%

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

       7. associate-/l/ 86.2%

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

       8. associate-/r* 86.2%

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

       9. *-un-lft-identity 86.2%

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

       10. unpow2 86.2%

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

       11. times-frac 89.1%

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

       12. *-commutative 89.1%

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

   11. Applied egg-rr 89.1%

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

       1. clear-num 89.1%

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

       2. un-div-inv 89.2%

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

       3. div-inv 89.1%

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

       4. clear-num 90.0%

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

       5. /-rgt-identity 90.0%

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

       6. *-commutative 90.0%

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

   13. Applied egg-rr 90.0%

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

4. Final simplification 78.2%

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

Alternative 10: 75.8% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (* z (* 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) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (z * (z * (y_m * x_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 * (z * (y_m * x_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 * (z * (y_m * x_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 * (z * (y_m * x_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(z * Float64(z * Float64(y_m * x_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 * (z * (y_m * x_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[(z * N[(z * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. remove-double-neg 92.9%

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

      2. distribute-lft-neg-out 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. associate-/r* 92.4%

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

      5. associate-/l/ 92.1%

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

      6. associate-/l/ 92.7%

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

      7. distribute-lft-neg-out 92.7%

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

      8. distribute-rgt-neg-in 92.7%

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

      9. distribute-lft-neg-in 92.7%

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

      10. remove-double-neg 92.7%

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

      11. sqr-neg 92.7%

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

      12. +-commutative 92.7%

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

      13. sqr-neg 92.7%

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

      14. fma-define 92.7%

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

      15. *-commutative 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   if 1 < z

   1. Initial program 87.5%

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

      1. remove-double-neg 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.3%

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

      5. associate-/l/ 87.3%

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

      6. associate-/l/ 88.2%

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

      7. distribute-lft-neg-out 88.2%

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

      8. distribute-rgt-neg-in 88.2%

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

      9. distribute-lft-neg-in 88.2%

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

      10. remove-double-neg 88.2%

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

      11. sqr-neg 88.2%

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

      12. +-commutative 88.2%

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

      13. sqr-neg 88.2%

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

      14. fma-define 88.2%

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

      15. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in z around inf 87.1%

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

      1. *-un-lft-identity 87.1%

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

      2. associate-/r* 87.1%

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

      3. pow-flip 87.2%

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

      4. metadata-eval 87.2%

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

      5. *-commutative 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. *-lft-identity 87.2%

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

      2. *-commutative 87.2%

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

   9. Simplified 87.2%

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

       1. div-inv 86.3%

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

       2. metadata-eval 86.3%

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

       3. pow-flip 86.2%

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

       4. associate-/r* 86.2%

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

       5. times-frac 86.4%

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

       6. *-un-lft-identity 86.4%

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

       7. associate-/l/ 86.2%

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

       8. associate-/r* 86.2%

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

       9. *-un-lft-identity 86.2%

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

       10. unpow2 86.2%

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

       11. times-frac 89.1%

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

       12. *-commutative 89.1%

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

   11. Applied egg-rr 89.1%

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

       1. *-commutative 89.1%

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

       2. clear-num 89.1%

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

       3. frac-times 89.1%

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

       4. metadata-eval 89.1%

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

       5. div-inv 89.1%

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

       6. clear-num 90.0%

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

       7. /-rgt-identity 90.0%

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

       8. *-commutative 90.0%

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

   13. Applied egg-rr 90.0%

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

4. Final simplification 78.2%

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

Alternative 11: 73.6% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 z) (* 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) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / ((z * z) * (y_m * x_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 * z) * (y_m * x_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 * z) * (y_m * x_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 * z) * (y_m * x_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(Float64(z * z) * Float64(y_m * x_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 * z) * (y_m * x_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[(N[(z * z), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. remove-double-neg 92.9%

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

      2. distribute-lft-neg-out 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. associate-/r* 92.4%

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

      5. associate-/l/ 92.1%

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

      6. associate-/l/ 92.7%

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

      7. distribute-lft-neg-out 92.7%

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

      8. distribute-rgt-neg-in 92.7%

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

      9. distribute-lft-neg-in 92.7%

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

      10. remove-double-neg 92.7%

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

      11. sqr-neg 92.7%

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

      12. +-commutative 92.7%

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

      13. sqr-neg 92.7%

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

      14. fma-define 92.7%

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

      15. *-commutative 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in z around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   if 1 < z

   1. Initial program 87.5%

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

      1. remove-double-neg 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.3%

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

      5. associate-/l/ 87.3%

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

      6. associate-/l/ 88.2%

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

      7. distribute-lft-neg-out 88.2%

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

      8. distribute-rgt-neg-in 88.2%

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

      9. distribute-lft-neg-in 88.2%

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

      10. remove-double-neg 88.2%

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

      11. sqr-neg 88.2%

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

      12. +-commutative 88.2%

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

      13. sqr-neg 88.2%

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

      14. fma-define 88.2%

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

      15. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in z around inf 87.1%

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

      1. unpow2 87.1%

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

   7. Applied egg-rr 87.1%

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

4. Final simplification 77.4%

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

Alternative 12: 58.3% 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 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 91.5%

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

   1. remove-double-neg 91.5%

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

   2. distribute-lft-neg-out 91.5%

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

   3. distribute-rgt-neg-in 91.5%

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

   4. associate-/r* 91.0%

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

   5. associate-/l/ 90.8%

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

   6. associate-/l/ 91.4%

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

   7. distribute-lft-neg-out 91.4%

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

   8. distribute-rgt-neg-in 91.4%

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

   9. distribute-lft-neg-in 91.4%

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

   10. remove-double-neg 91.4%

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

   11. sqr-neg 91.4%

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

   12. +-commutative 91.4%

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

   13. sqr-neg 91.4%

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

   14. fma-define 91.5%

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

   15. *-commutative 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in z around 0 58.8%

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

   1. *-commutative 58.8%

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

   2. associate-/r* 58.9%

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

7. Simplified 58.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]
8. Add Preprocessing

Alternative 13: 58.3% 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 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 91.5%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0 58.9%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]
4. Add Preprocessing

Alternative 14: 58.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \frac{1}{y\_m \cdot x\_m}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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(1.0 / Float64(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[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. remove-double-neg 91.5%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{-\left(-y \cdot \left(1 + z \cdot z\right)\right)}} \]

   2. distribute-lft-neg-out 91.5%

      \[\leadsto \frac{\frac{1}{x}}{-\color{blue}{\left(-y\right) \cdot \left(1 + z \cdot z\right)}} \]

   3. distribute-rgt-neg-in 91.5%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(-y\right) \cdot \left(-\left(1 + z \cdot z\right)\right)}} \]

   4. associate-/r* 91.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{-y}}{-\left(1 + z \cdot z\right)}} \]

   5. associate-/l/ 90.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{\left(-y\right) \cdot x}}}{-\left(1 + z \cdot z\right)} \]

   6. associate-/l/ 91.4%

      \[\leadsto \color{blue}{\frac{1}{\left(-\left(1 + z \cdot z\right)\right) \cdot \left(\left(-y\right) \cdot x\right)}} \]

   7. distribute-lft-neg-out 91.4%

      \[\leadsto \frac{1}{\left(-\left(1 + z \cdot z\right)\right) \cdot \color{blue}{\left(-y \cdot x\right)}} \]

   8. distribute-rgt-neg-in 91.4%

      \[\leadsto \frac{1}{\color{blue}{-\left(-\left(1 + z \cdot z\right)\right) \cdot \left(y \cdot x\right)}} \]

   9. distribute-lft-neg-in 91.4%

      \[\leadsto \frac{1}{\color{blue}{\left(-\left(-\left(1 + z \cdot z\right)\right)\right) \cdot \left(y \cdot x\right)}} \]

   10. remove-double-neg 91.4%

       \[\leadsto \frac{1}{\color{blue}{\left(1 + z \cdot z\right)} \cdot \left(y \cdot x\right)} \]

   11. sqr-neg 91.4%

       \[\leadsto \frac{1}{\left(1 + \color{blue}{\left(-z\right) \cdot \left(-z\right)}\right) \cdot \left(y \cdot x\right)} \]

   12. +-commutative 91.4%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(-z\right) \cdot \left(-z\right) + 1\right)} \cdot \left(y \cdot x\right)} \]

   13. sqr-neg 91.4%

       \[\leadsto \frac{1}{\left(\color{blue}{z \cdot z} + 1\right) \cdot \left(y \cdot x\right)} \]

   14. fma-define 91.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(y \cdot x\right)} \]

   15. *-commutative 91.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \color{blue}{\left(x \cdot y\right)}} \]

3. Simplified 91.5%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x \cdot y\right)}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 58.8%

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]

6. Final simplification 58.8%

   \[\leadsto \frac{1}{y \cdot x} \]
7. Add Preprocessing

Developer Target 1: 92.2% accurate, 0.3× 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 2024160 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))