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

Percentage Accurate: 89.8% → 98.9%

Time: 10.9s

Alternatives: 6

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% 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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x\_m}}{y}}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (*
    x_s
    (if (<= t_0 2e+306) (/ (/ 1.0 x_m) t_0) (/ (/ (/ (/ 1.0 z) x_m) y) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (((1.0 / z) / x_m) / y) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 2d+306) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (((1.0d0 / z) / x_m) / y) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (((1.0 / z) / x_m) / y) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+306:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (((1.0 / z) / x_m) / y) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+306)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / x_m) / y) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+306)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (((1.0 / z) / x_m) / y) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e+306], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 2.00000000000000003e306

   1. Initial program 94.9%

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

   if 2.00000000000000003e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 72.0%

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

      1. associate-/l/ 72.0%

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

      2. associate-*l* 78.7%

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

      3. *-commutative 78.7%

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

      4. sqr-neg 78.7%

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

      5. +-commutative 78.7%

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

      6. sqr-neg 78.7%

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

      7. fma-define 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in z around inf 72.0%

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

      1. associate-*r* 78.4%

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

      2. associate-/r* 79.5%

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

   7. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. associate-/l/ 78.4%

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

      3. associate-/r* 79.6%

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

      4. pow-flip 80.0%

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

      5. metadata-eval 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-lft-identity 80.0%

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

   11. Simplified 80.0%

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

       1. sqr-pow 80.0%

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

       2. associate-/l* 90.7%

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

       3. metadata-eval 90.7%

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

       4. unpow-1 90.7%

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

       5. metadata-eval 90.7%

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

       6. unpow-1 90.7%

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

   13. Applied egg-rr 90.7%

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

       1. associate-*l/ 90.7%

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

       2. *-un-lft-identity 90.7%

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

       3. associate-/r* 99.9%

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

   15. Applied egg-rr 99.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \frac{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x\_m}\right)}^{-2}}{y} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (/ (pow (* (hypot 1.0 z) (sqrt x_m)) -2.0) y)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (pow((hypot(1.0, z) * sqrt(x_m)), -2.0) / y);
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (Math.pow((Math.hypot(1.0, z) * Math.sqrt(x_m)), -2.0) / y);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	return x_s * (math.pow((math.hypot(1.0, z) * math.sqrt(x_m)), -2.0) / y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64((Float64(hypot(1.0, z) * sqrt(x_m)) ^ -2.0) / y))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (((hypot(1.0, z) * sqrt(x_m)) ^ -2.0) / y);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[Power[N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 91.1%

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

   3. *-commutative 91.1%

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

   4. sqr-neg 91.1%

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

   5. +-commutative 91.1%

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

   6. sqr-neg 91.1%

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

   7. fma-define 91.1%

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

3. Simplified 91.1%

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

   1. associate-*r* 92.2%

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

   2. *-commutative 92.2%

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

   3. associate-/r* 92.6%

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

   4. *-commutative 92.6%

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

   5. associate-/l/ 92.6%

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

   6. fma-undefine 92.6%

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

   7. +-commutative 92.6%

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

   8. associate-/r* 91.1%

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

   9. *-un-lft-identity 91.1%

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

   10. add-sqr-sqrt 46.7%

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

   11. times-frac 46.7%

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

   12. +-commutative 46.7%

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

   13. fma-undefine 46.7%

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

   14. *-commutative 46.7%

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

   15. sqrt-prod 46.7%

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

   16. fma-undefine 46.7%

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

   17. +-commutative 46.7%

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

   18. hypot-1-def 46.7%

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

   19. +-commutative 46.7%

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

6. Applied egg-rr 51.2%

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

   1. associate-*l/ 51.2%

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

   2. *-lft-identity 51.2%

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

   3. associate-/l/ 51.3%

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

   4. *-commutative 51.3%

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

8. Simplified 51.3%

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

   1. div-inv 51.3%

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

   2. associate-/r* 51.2%

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

   3. frac-times 48.8%

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

   4. swap-sqr 46.7%

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

   5. hypot-undefine 46.7%

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

   6. metadata-eval 46.7%

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

   7. unpow2 46.7%

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

   8. hypot-undefine 46.6%

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

   9. metadata-eval 46.6%

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

   10. unpow2 46.6%

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

   11. add-sqr-sqrt 46.7%

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

   12. +-commutative 46.7%

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

   13. unpow2 46.7%

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

   14. fma-undefine 46.7%

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

   15. add-sqr-sqrt 91.1%

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

   16. frac-times 91.9%

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

10. Applied egg-rr 46.9%

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

    1. associate-/r* 47.0%

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

    2. associate-/l/ 46.9%

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

    3. associate-*r/ 45.3%

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

    4. unpow-1 45.3%

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

    5. unpow-1 45.3%

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

    6. pow-sqr 45.3%

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

    7. metadata-eval 45.3%

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

12. Simplified 45.3%

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

13. Final simplification 45.3%

    \[\leadsto \frac{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x}\right)}^{-2}}{y} \]
14. Add Preprocessing

Alternative 3: 76.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0 70.9%

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

   if 1 < z

   1. Initial program 83.1%

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

      1. associate-/l/ 83.2%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. sqr-neg 81.6%

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

      5. +-commutative 81.6%

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

      6. sqr-neg 81.6%

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

      7. fma-define 81.6%

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

   3. Simplified 81.6%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

      3. *-commutative 84.3%

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

      4. add-sqr-sqrt 33.7%

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

      5. pow2 33.7%

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

      6. sqrt-prod 33.8%

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

      7. fma-undefine 33.8%

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

      8. +-commutative 33.8%

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

      9. hypot-1-def 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in z around inf 35.4%

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

      1. unpow2 35.4%

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

      2. swap-sqr 33.7%

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

      3. add-sqr-sqrt 82.1%

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

      4. associate-*r* 86.9%

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

   9. Applied egg-rr 86.9%

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

4. Final simplification 74.8%

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

Alternative 4: 78.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0 70.9%

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

   if 1 < z

   1. Initial program 83.1%

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

      1. associate-/l/ 83.2%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. sqr-neg 81.6%

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

      5. +-commutative 81.6%

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

      6. sqr-neg 81.6%

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

      7. fma-define 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in z around inf 81.0%

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

      1. associate-*r* 82.1%

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

      2. associate-/r* 83.9%

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

   7. Simplified 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. associate-/l/ 82.1%

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

      3. associate-/r* 83.9%

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

      4. pow-flip 84.0%

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

      5. metadata-eval 84.0%

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

   9. Applied egg-rr 84.0%

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

       1. *-lft-identity 84.0%

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

   11. Simplified 84.0%

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

       1. sqr-pow 83.9%

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

       2. associate-/l* 89.7%

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

       3. metadata-eval 89.7%

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

       4. unpow-1 89.7%

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

       5. metadata-eval 89.7%

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

       6. unpow-1 89.7%

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

   13. Applied egg-rr 89.7%

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

       1. associate-/l/ 89.6%

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

       2. un-div-inv 89.6%

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

       3. *-commutative 89.6%

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

       4. associate-*r* 95.9%

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

   15. Applied egg-rr 95.9%

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

4. Final simplification 77.0%

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

Alternative 5: 78.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0 70.9%

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

   if 1 < z

   1. Initial program 83.1%

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

      1. associate-/l/ 83.2%

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

      2. associate-*l* 81.6%

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

      3. *-commutative 81.6%

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

      4. sqr-neg 81.6%

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

      5. +-commutative 81.6%

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

      6. sqr-neg 81.6%

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

      7. fma-define 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in z around inf 81.0%

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

      1. associate-*r* 82.1%

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

      2. associate-/r* 83.9%

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

   7. Simplified 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. associate-/l/ 82.1%

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

      3. associate-/r* 83.9%

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

      4. pow-flip 84.0%

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

      5. metadata-eval 84.0%

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

   9. Applied egg-rr 84.0%

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

       1. *-lft-identity 84.0%

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

   11. Simplified 84.0%

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

       1. sqr-pow 83.9%

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

       2. associate-/l* 89.7%

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

       3. metadata-eval 89.7%

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

       4. unpow-1 89.7%

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

       5. metadata-eval 89.7%

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

       6. unpow-1 89.7%

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

   13. Applied egg-rr 89.7%

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

       1. associate-*l/ 89.8%

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

       2. *-un-lft-identity 89.8%

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

       3. associate-/r* 96.0%

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

   15. Applied egg-rr 96.0%

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

4. Final simplification 77.0%

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

Alternative 6: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 91.1%

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

   3. *-commutative 91.1%

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

   4. sqr-neg 91.1%

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

   5. +-commutative 91.1%

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

   6. sqr-neg 91.1%

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

   7. fma-define 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in z around 0 58.7%

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

6. Final simplification 58.7%

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

Developer target: 92.7% 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 2024079 
(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)))))