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

Percentage Accurate: 88.6% → 98.5%

Time: 9.0s

Alternatives: 6

Speedup: 1.3×

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: 88.6% 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: 98.5% accurate, 0.3× speedup

Math

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

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 (/ (pow (fma (* x_m z) z x_m) -1.0) y_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) {
	return x_s * (y_s * (pow(fma((x_m * z), z, x_m), -1.0) / y_m));
}

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)
	return Float64(x_s * Float64(y_s * Float64((fma(Float64(x_m * z), z, x_m) ^ -1.0) / y_m)))
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 * N[(N[Power[N[(N[(x$95$m * z), $MachinePrecision] * z + x$95$m), $MachinePrecision], -1.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-frac-neg N/A

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

   5. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}}{y \cdot \left(1 + z \cdot z\right)} \]

   6. div-sub N/A

      \[\leadsto \color{blue}{\frac{0}{y \cdot \left(1 + z \cdot z\right)} - \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y \cdot \left(1 + z \cdot z\right)}} \]

   7. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)}} - \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y \cdot \left(1 + z \cdot z\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)} - \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{y \cdot \left(1 + z \cdot z\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{0}{\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)} - \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\color{blue}{y \cdot \left(1 + z \cdot z\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{0}{\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)} - \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\color{blue}{\left(1 + z \cdot z\right) \cdot y}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{0}{\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)} - \color{blue}{\frac{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{1 + z \cdot z}}{y}} \]

   12. frac-sub N/A

       \[\leadsto \color{blue}{\frac{0 \cdot y - \left(\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{1 + z \cdot z}}{\left(\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)\right) \cdot y}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{0 \cdot y - \left(\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)\right) \cdot \frac{\mathsf{neg}\left(\frac{1}{x}\right)}{1 + z \cdot z}}{\left(\mathsf{neg}\left(y \cdot \left(1 + z \cdot z\right)\right)\right) \cdot y}} \]

4. Applied rewrites 42.5%

   \[\leadsto \color{blue}{\frac{0 \cdot y - \left(\left(-y\right) \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot \frac{\frac{-1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{\left(\left(-y\right) \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot y}} \]

6. Applied rewrites 93.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow-1 N/A

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

   5. un-div-inv N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. distribute-lft-in N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-/.f64 N/A

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

8. Applied rewrites 93.5%

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

9. Final simplification 93.5%

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

Alternative 2: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\_m\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

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) 5e-6)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* (* z z) x_m) y_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) <= 5e-6) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / (((z * z) * x_m) * y_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) <= 5e-6)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * x_m) * y_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], 5e-6], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 80.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 80.3

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

   4. Applied rewrites 80.3%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.5

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

   9. Applied rewrites 78.5%

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

Alternative 3: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

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) 5e-6)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* z 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) <= 5e-6) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / ((z * 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) <= 5e-6)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(z * 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], 5e-6], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 78.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x\_m}\\ \end{array}\right) \end{array} \]

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) 5e-6)
     (/ (fma (- z) z 1.0) (* y_m x_m))
     (/ 1.0 (* (* (* z 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) <= 5e-6) {
		tmp = fma(-z, z, 1.0) / (y_m * x_m);
	} else {
		tmp = 1.0 / (((z * 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) <= 5e-6)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * 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], 5e-6], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (*.f64 z z)

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.3× speedup

Math

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

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 (/ 1.0 (* (fma (* x_m z) z x_m) y_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) {
	return x_s * (y_s * (1.0 / (fma((x_m * z), z, x_m) * y_m)));
}

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)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(fma(Float64(x_m * z), z, x_m) * y_m))))
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 * N[(1.0 / N[(N[(N[(x$95$m * z), $MachinePrecision] * z + x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 89.2

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 89.2

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

4. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft1-in N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 96.6

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 96.6

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

6. Applied rewrites 96.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 93.3

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.3

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

8. Applied rewrites 93.3%

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

9. Final simplification 93.3%

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

Alternative 6: 59.1% accurate, 2.1× speedup

Math

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

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 (/ 1.0 (* 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) {
	return x_s * (y_s * (1.0 / (y_m * x_m)));
}

Fortran

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

Java

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

Python

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

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)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (1.0 / (y_m * x_m)));
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 * N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 52.7

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

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\frac{1}{y \cdot x}} \]
6. Add Preprocessing

Developer Target 1: 92.8% 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 2024243 
(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)))))