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

Percentage Accurate: 89.4% → 98.4%

Time: 11.2s

Alternatives: 11

Speedup: 1.1×

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: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, x\_m \cdot z, y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 4e+214)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ 1.0 (fma (* y_m z) (* x_m z) (* y_m x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 4e+214) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = 1.0 / fma((y_m * z), (x_m * z), (y_m * x_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+214)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(x_m * z), Float64(y_m * x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 4e+214], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision] + N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e214

   1. Initial program 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. /-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if 3.9999999999999998e214 < (*.f64 z z)

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.5

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 70.5

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

   4. Applied egg-rr 70.5%

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

   6. Applied egg-rr 98.6%

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

Alternative 2: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\frac{1}{y\_m \cdot \mathsf{fma}\left(x\_m, z \cdot z, x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\left(y\_m \cdot z\right) \cdot \left(x\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 1e+192)
     (/ 1.0 (* y_m (fma x_m (* z z) x_m)))
     (/ (- -1.0) (* (* y_m z) (* x_m z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+192) {
		tmp = 1.0 / (y_m * fma(x_m, (z * z), x_m));
	} else {
		tmp = -(-1.0) / ((y_m * z) * (x_m * z));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+192)
		tmp = Float64(1.0 / Float64(y_m * fma(x_m, Float64(z * z), x_m)));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(Float64(y_m * z) * Float64(x_m * z)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+192], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(y$95$m * z), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e192

   1. Initial program 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.6

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 96.6

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

   4. Applied egg-rr 96.6%

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

   6. Applied egg-rr 99.2%

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

   if 1.00000000000000004e192 < (*.f64 z z)

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. /-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 81.0

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

   4. Applied egg-rr 81.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 81.0

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

   7. Simplified 81.0%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. remove-double-div N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. frac-2neg N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. distribute-rgt-neg-in N/A

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

      23. lower-*.f64 N/A

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

      24. lower-neg.f64 96.0

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

   9. Applied egg-rr 96.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+192}:\\ \;\;\;\;\frac{1}{y \cdot \mathsf{fma}\left(x, z \cdot z, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 93.1% accurate, 0.5× 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 2024218 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

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