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

Percentage Accurate: 88.7% → 98.4%

Time: 12.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y\_m}}{z\_m}}{\mathsf{hypot}\left(1, z\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+169)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (/ (/ (/ 1.0 y_m) z_m) (* (hypot 1.0 z_m) x_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+169) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = ((1.0 / y_m) / z_m) / (hypot(1.0, z_m) * x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+169)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z_m) / Float64(hypot(1.0, z_m) * x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+169], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999987e169

   1. Initial program 96.8%

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

      1. associate-/l/ 96.5%

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

      2. associate-*l* 98.1%

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

      3. *-commutative 98.1%

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

      4. sqr-neg 98.1%

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

      5. +-commutative 98.1%

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

      6. sqr-neg 98.1%

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

      7. fma-define 98.1%

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

   3. Simplified 98.1%

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

   if 1.99999999999999987e169 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. associate-/l/ 81.0%

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

      2. associate-*l* 82.1%

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

      3. *-commutative 82.1%

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

      4. sqr-neg 82.1%

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

      5. +-commutative 82.1%

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

      6. sqr-neg 82.1%

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

      7. fma-define 82.1%

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

   3. Simplified 82.1%

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

      1. associate-*r* 80.7%

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

      2. *-commutative 80.7%

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

      3. associate-/r* 79.6%

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

      4. *-commutative 79.6%

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

      5. associate-/l/ 79.6%

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

      6. fma-undefine 79.6%

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

      7. +-commutative 79.6%

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

      8. associate-/r* 80.9%

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

      9. *-un-lft-identity 80.9%

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

      10. add-sqr-sqrt 41.8%

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

      11. times-frac 41.8%

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

      12. +-commutative 41.8%

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

      13. fma-undefine 41.8%

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

      14. *-commutative 41.8%

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

      15. sqrt-prod 41.9%

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

      16. fma-undefine 41.9%

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

      17. +-commutative 41.9%

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

      18. hypot-1-def 41.9%

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

      19. +-commutative 41.9%

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

   6. Applied egg-rr 54.0%

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

      1. associate-/l/ 53.9%

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

      2. associate-*r/ 53.9%

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

      3. *-rgt-identity 53.9%

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

      4. *-commutative 53.9%

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

      5. associate-/r* 53.9%

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

      6. *-commutative 53.9%

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

   8. Simplified 53.9%

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

      1. div-inv 53.9%

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

      2. div-inv 53.9%

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

      3. associate-*l* 53.9%

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

      4. pow1/2 53.9%

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

      5. pow-flip 54.0%

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

      6. metadata-eval 54.0%

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

      7. associate-/l/ 54.0%

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

      8. associate-/l/ 54.0%

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

      9. associate-/l/ 54.0%

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

      10. pow1/2 54.0%

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

      11. pow-flip 53.9%

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

      12. metadata-eval 53.9%

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

   10. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{{y}^{-0.5} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 54.0%

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

       2. associate-*l/ 54.0%

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

       3. *-lft-identity 54.0%

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

       4. associate-*r/ 53.9%

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

       5. associate-*r/ 54.0%

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

       6. pow-sqr 99.8%

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

       7. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in z around inf 84.0%

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

       1. associate-/r* 83.9%

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

   15. Simplified 83.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{\mathsf{hypot}\left(1, z\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.3%

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

   2. associate-*l* 92.8%

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

   3. *-commutative 92.8%

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

   4. sqr-neg 92.8%

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

   5. +-commutative 92.8%

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

   6. sqr-neg 92.8%

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

   7. fma-define 92.8%

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

3. Simplified 92.8%

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

   1. associate-*r* 92.3%

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

   2. *-commutative 92.3%

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

   3. associate-/r* 92.0%

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

   4. *-commutative 92.0%

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

   5. associate-/l/ 92.3%

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

   6. fma-undefine 92.3%

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

   7. +-commutative 92.3%

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

   8. associate-/r* 91.6%

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

   9. *-un-lft-identity 91.6%

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

   10. add-sqr-sqrt 48.0%

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

   11. times-frac 48.0%

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

   12. +-commutative 48.0%

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

   13. fma-undefine 48.0%

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

   14. *-commutative 48.0%

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

   15. sqrt-prod 48.0%

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

   16. fma-undefine 48.0%

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

   17. +-commutative 48.0%

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

   18. hypot-1-def 48.0%

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

   19. +-commutative 48.0%

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

6. Applied egg-rr 52.8%

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

   1. associate-/l/ 52.8%

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

   2. associate-*r/ 52.8%

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

   3. *-rgt-identity 52.8%

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

   4. *-commutative 52.8%

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

   5. associate-/r* 52.8%

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

   6. *-commutative 52.8%

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

8. Simplified 52.8%

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

9. Final simplification 52.8%

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

Alternative 3: 98.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.3%

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

   2. associate-*l* 92.8%

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

   3. *-commutative 92.8%

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

   4. sqr-neg 92.8%

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

   5. +-commutative 92.8%

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

   6. sqr-neg 92.8%

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

   7. fma-define 92.8%

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

3. Simplified 92.8%

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

   1. associate-*r* 92.3%

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

   2. *-commutative 92.3%

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

   3. associate-/r* 92.0%

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

   4. *-commutative 92.0%

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

   5. associate-/l/ 92.3%

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

   6. fma-undefine 92.3%

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

   7. +-commutative 92.3%

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

   8. associate-/r* 91.6%

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

   9. *-un-lft-identity 91.6%

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

   10. add-sqr-sqrt 48.0%

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

   11. times-frac 48.0%

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

   12. +-commutative 48.0%

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

   13. fma-undefine 48.0%

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

   14. *-commutative 48.0%

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

   15. sqrt-prod 48.0%

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

   16. fma-undefine 48.0%

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

   17. +-commutative 48.0%

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

   18. hypot-1-def 48.0%

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

   19. +-commutative 48.0%

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

6. Applied egg-rr 52.8%

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

   1. associate-/l/ 52.8%

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

   2. associate-*r/ 52.8%

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

   3. *-rgt-identity 52.8%

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

   4. *-commutative 52.8%

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

   5. associate-/r* 52.8%

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

   6. *-commutative 52.8%

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

8. Simplified 52.8%

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

   1. div-inv 52.8%

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

   2. div-inv 52.8%

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

   3. associate-*l* 52.4%

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

   4. pow1/2 52.4%

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

   5. pow-flip 52.5%

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

   6. metadata-eval 52.5%

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

   7. associate-/l/ 52.5%

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

   8. associate-/l/ 52.5%

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

   9. associate-/l/ 52.2%

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

   10. pow1/2 52.2%

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

   11. pow-flip 52.1%

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

   12. metadata-eval 52.1%

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

10. Applied egg-rr 52.1%

    \[\leadsto \color{blue}{{y}^{-0.5} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 52.1%

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

    2. associate-*l/ 52.1%

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

    3. *-lft-identity 52.1%

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

    4. associate-*r/ 52.1%

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

    5. associate-*r/ 52.1%

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

    6. pow-sqr 98.4%

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

    7. metadata-eval 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

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

Alternative 4: 98.0% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot 1.0 z_m))))
   (* y_s (* x_s (* (/ t_0 y_m) (/ t_0 x_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 / hypot(1.0, z_m);
	return y_s * (x_s * ((t_0 / y_m) * (t_0 / x_m)));
}

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = 1.0 / Math.hypot(1.0, z_m);
	return y_s * (x_s * ((t_0 / y_m) * (t_0 / x_m)));
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = 1.0 / math.hypot(1.0, z_m)
	return y_s * (x_s * ((t_0 / y_m) * (t_0 / x_m)))

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(1.0 / hypot(1.0, z_m))
	return Float64(y_s * Float64(x_s * Float64(Float64(t_0 / y_m) * Float64(t_0 / x_m))))
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = 1.0 / hypot(1.0, z_m);
	tmp = y_s * (x_s * ((t_0 / y_m) * (t_0 / x_m)));
end

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.3%

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

   2. associate-*l* 92.8%

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

   3. *-commutative 92.8%

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

   4. sqr-neg 92.8%

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

   5. +-commutative 92.8%

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

   6. sqr-neg 92.8%

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

   7. fma-define 92.8%

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

3. Simplified 92.8%

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

   1. clear-num 92.8%

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

   2. associate-*r* 92.3%

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

   3. *-commutative 92.3%

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

   4. *-commutative 92.3%

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

   5. associate-/r/ 92.3%

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

   6. associate-/r* 92.3%

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

6. Applied egg-rr 92.3%

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

   1. add-sqr-sqrt 92.3%

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

   2. *-commutative 92.3%

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

   3. times-frac 93.0%

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

   4. clear-num 93.0%

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

   5. sqrt-div 93.0%

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

   6. metadata-eval 93.0%

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

   7. /-rgt-identity 93.0%

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

   8. fma-undefine 93.0%

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

   9. unpow2 93.0%

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

   10. +-commutative 93.0%

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

   11. metadata-eval 93.0%

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

   12. unpow2 93.0%

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

   13. hypot-undefine 93.0%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 5: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m}}{y\_m} \cdot \frac{\frac{1}{z\_m}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+307)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (* (/ (/ 1.0 z_m) y_m) (/ (/ 1.0 z_m) x_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+307) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+307)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y_m) * Float64(Float64(1.0 / z_m) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+307], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 96.2%

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

      1. associate-/l/ 95.8%

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

      2. associate-*l* 97.8%

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

      3. *-commutative 97.8%

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

      4. sqr-neg 97.8%

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

      5. +-commutative 97.8%

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

      6. sqr-neg 97.8%

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

      7. fma-define 97.8%

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

   3. Simplified 97.8%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 77.4%

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

      1. associate-/l/ 77.4%

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

      2. associate-*l* 77.4%

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

      3. *-commutative 77.4%

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

      4. sqr-neg 77.4%

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

      5. +-commutative 77.4%

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

      6. sqr-neg 77.4%

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

      7. fma-define 77.4%

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

   3. Simplified 77.4%

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

      1. clear-num 77.4%

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

      2. associate-*r* 77.1%

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

      3. *-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-/r/ 77.1%

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

      6. associate-/r* 77.1%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in z around inf 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. *-commutative 77.1%

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

      3. times-frac 77.4%

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

      4. sqrt-div 77.4%

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

      5. metadata-eval 77.4%

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

      6. sqrt-pow1 77.4%

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

      7. metadata-eval 77.4%

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

      8. pow1 77.4%

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

      9. sqrt-div 77.4%

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

      10. metadata-eval 77.4%

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

      11. sqrt-pow1 99.8%

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

      12. metadata-eval 99.8%

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

      13. pow1 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y} \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 0.5× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 x_m) t_0)
       (* (/ 1.0 z_m) (/ (/ 1.0 z_m) (* y_m x_m))))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / z_m) / Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / z_m) * ((1.0 / z_m) / (y_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < +inf.0

   1. Initial program 91.6%

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

   if +inf.0 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 91.6%

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

      1. associate-/l/ 91.3%

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

      2. associate-*l* 92.8%

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

      3. *-commutative 92.8%

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

      4. sqr-neg 92.8%

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

      5. +-commutative 92.8%

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

      6. sqr-neg 92.8%

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

      7. fma-define 92.8%

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

   3. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. associate-*r* 92.3%

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

      3. *-commutative 92.3%

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

      4. *-commutative 92.3%

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

      5. associate-/r/ 92.3%

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

      6. associate-/r* 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in z around inf 50.2%

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

      1. add-sqr-sqrt 50.2%

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

      2. associate-/l* 50.2%

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

      3. sqrt-div 50.2%

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

      4. metadata-eval 50.2%

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

      5. sqrt-pow1 39.5%

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

      6. metadata-eval 39.5%

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

      7. pow1 39.5%

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

      8. sqrt-div 39.5%

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

      9. metadata-eval 39.5%

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

      10. sqrt-pow1 56.7%

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

      11. metadata-eval 56.7%

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

      12. pow1 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 91.6%

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

Alternative 7: 88.7% accurate, 0.5× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       (/ (/ 1.0 x_m) t_0)
       (* (/ (/ 1.0 z_m) y_m) (/ (/ 1.0 z_m) x_m)))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x_m);
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x_m);
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x_m)
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y_m) * Float64(Float64(1.0 / z_m) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < +inf.0

   1. Initial program 91.6%

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

   if +inf.0 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 91.6%

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

      1. associate-/l/ 91.3%

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

      2. associate-*l* 92.8%

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

      3. *-commutative 92.8%

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

      4. sqr-neg 92.8%

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

      5. +-commutative 92.8%

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

      6. sqr-neg 92.8%

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

      7. fma-define 92.8%

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

   3. Simplified 92.8%

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

      1. clear-num 92.8%

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

      2. associate-*r* 92.3%

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

      3. *-commutative 92.3%

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

      4. *-commutative 92.3%

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

      5. associate-/r/ 92.3%

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

      6. associate-/r* 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in z around inf 50.2%

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

      1. add-sqr-sqrt 50.2%

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

      2. *-commutative 50.2%

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

      3. times-frac 49.6%

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

      4. sqrt-div 49.6%

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

      5. metadata-eval 49.6%

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

      6. sqrt-pow1 39.0%

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

      7. metadata-eval 39.0%

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

      8. pow1 39.0%

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

      9. sqrt-div 39.0%

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

      10. metadata-eval 39.0%

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

      11. sqrt-pow1 55.1%

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

      12. metadata-eval 55.1%

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

      13. pow1 55.1%

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

   9. Applied egg-rr 55.1%

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

4. Final simplification 91.6%

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

Alternative 8: 73.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.6%

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

      1. associate-/l/ 93.3%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

      4. sqr-neg 93.8%

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

      5. +-commutative 93.8%

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

      6. sqr-neg 93.8%

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

      7. fma-define 93.8%

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

   3. Simplified 93.8%

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

      1. associate-*r* 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-/r* 93.7%

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

      4. *-commutative 93.7%

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

      5. associate-/l/ 94.1%

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

      6. fma-undefine 94.1%

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

      7. +-commutative 94.1%

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

      8. associate-/r* 93.6%

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

      9. *-un-lft-identity 93.6%

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

      10. add-sqr-sqrt 50.9%

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

      11. times-frac 50.9%

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

      12. +-commutative 50.9%

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

      13. fma-undefine 50.9%

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

      14. *-commutative 50.9%

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

      15. sqrt-prod 50.9%

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

      16. fma-undefine 50.9%

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

      17. +-commutative 50.9%

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

      18. hypot-1-def 50.9%

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

      19. +-commutative 50.9%

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

   6. Applied egg-rr 54.4%

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

      1. associate-/l/ 54.5%

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

      2. associate-*r/ 54.4%

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

      3. *-rgt-identity 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-/r* 54.4%

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

      6. *-commutative 54.4%

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

   8. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. div-inv 54.5%

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

      3. associate-*l* 54.0%

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

      4. pow1/2 54.0%

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

      5. pow-flip 54.0%

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

      6. metadata-eval 54.0%

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

      7. associate-/l/ 54.0%

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

      8. associate-/l/ 54.0%

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

      9. associate-/l/ 53.5%

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

      10. pow1/2 53.5%

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

      11. pow-flip 53.5%

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

      12. metadata-eval 53.5%

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

   10. Applied egg-rr 53.5%

       \[\leadsto \color{blue}{{y}^{-0.5} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 53.5%

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

       2. associate-*l/ 53.5%

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

       3. *-lft-identity 53.5%

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

       4. associate-*r/ 53.5%

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

       5. associate-*r/ 53.5%

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

       6. pow-sqr 98.2%

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

       7. metadata-eval 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in z around 0 74.5%

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

       1. associate-/l/ 74.9%

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

   15. Simplified 74.9%

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

   if 1 < z

   1. Initial program 85.8%

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

      1. associate-/l/ 85.9%

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

      2. associate-*l* 90.2%

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

      3. *-commutative 90.2%

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

      4. sqr-neg 90.2%

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

      5. +-commutative 90.2%

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

      6. sqr-neg 90.2%

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

      7. fma-define 90.2%

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

   3. Simplified 90.2%

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

      1. associate-*r* 88.6%

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

      2. *-commutative 88.6%

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

      3. associate-/r* 87.2%

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

      4. *-commutative 87.2%

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

      5. associate-/l/ 87.3%

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

      6. fma-undefine 87.3%

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

      7. +-commutative 87.3%

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

      8. associate-/r* 85.8%

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

      9. *-un-lft-identity 85.8%

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

      10. add-sqr-sqrt 40.0%

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

      11. times-frac 40.0%

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

      12. +-commutative 40.0%

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

      13. fma-undefine 40.0%

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

      14. *-commutative 40.0%

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

      15. sqrt-prod 40.1%

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

      16. fma-undefine 40.1%

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

      17. +-commutative 40.1%

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

      18. hypot-1-def 40.1%

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

      19. +-commutative 40.1%

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

   6. Applied egg-rr 48.2%

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

      1. associate-/l/ 48.2%

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

      2. associate-*r/ 48.2%

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

      3. *-rgt-identity 48.2%

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

      4. *-commutative 48.2%

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

      5. associate-/r* 48.3%

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

      6. *-commutative 48.3%

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

   8. Simplified 48.3%

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

      1. div-inv 48.3%

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

      2. div-inv 48.2%

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

      3. associate-*l* 48.1%

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

      4. pow1/2 48.1%

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

      5. pow-flip 48.2%

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

      6. metadata-eval 48.2%

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

      7. associate-/l/ 48.3%

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

      8. associate-/l/ 48.3%

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

      9. associate-/l/ 48.3%

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

      10. pow1/2 48.3%

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

      11. pow-flip 48.2%

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

      12. metadata-eval 48.2%

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

   10. Applied egg-rr 48.2%

       \[\leadsto \color{blue}{{y}^{-0.5} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 48.2%

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

       2. associate-*l/ 48.2%

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

       3. *-lft-identity 48.2%

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

       4. associate-*r/ 48.3%

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

       5. associate-*r/ 48.3%

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

       6. pow-sqr 99.2%

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

       7. metadata-eval 99.2%

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

   12. Simplified 99.2%

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

   13. Taylor expanded in z around 0 47.7%

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

   14. Taylor expanded in z around inf 46.6%

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

4. Final simplification 67.3%

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

Alternative 9: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Final simplification 91.6%

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

Alternative 10: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/l/ 91.3%

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

   2. associate-*l* 92.8%

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

   3. *-commutative 92.8%

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

   4. sqr-neg 92.8%

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

   5. +-commutative 92.8%

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

   6. sqr-neg 92.8%

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

   7. fma-define 92.8%

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

3. Simplified 92.8%

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

5. Taylor expanded in z around 0 61.6%

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

6. Final simplification 61.6%

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

Developer target: 92.8% 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 2024096 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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