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

Percentage Accurate: 90.4% → 97.6%

Time: 9.1s

Alternatives: 11

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: 90.4% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot z, x\_m, x\_m\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x\_m \cdot \left(z \cdot y\right), z, x\_m \cdot y\right)}\\ \end{array} \end{array} \]

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 z) 5e+282)
    (/ 1.0 (* (fma (* z z) x_m x_m) y))
    (/ 1.0 (fma (* x_m (* z y)) 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 * z) <= 5e+282) {
		tmp = 1.0 / (fma((z * z), x_m, x_m) * y);
	} else {
		tmp = 1.0 / fma((x_m * (z * y)), 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 (Float64(z * z) <= 5e+282)
		tmp = Float64(1.0 / Float64(fma(Float64(z * z), x_m, x_m) * y));
	else
		tmp = Float64(1.0 / fma(Float64(x_m * Float64(z * y)), z, Float64(x_m * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[N[(z * z), $MachinePrecision], 5e+282], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision] * z + N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999978e282

   1. Initial program 97.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 97.4

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

      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 97.4

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

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 97.4

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

   9. Applied rewrites 97.4%

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

   if 4.99999999999999978e282 < (*.f64 z z)

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      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 69.5

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

      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 69.5

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

   4. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot z, x, x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot \left(z \cdot y\right), z, x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 z) 4e-8)
    (/ (fma (- z) z 1.0) (* x_m y))
    (/ 1.0 (* (* (* z z) y) 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 * z) <= 4e-8) {
		tmp = fma(-z, z, 1.0) / (x_m * y);
	} else {
		tmp = 1.0 / (((z * z) * y) * 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 (Float64(z * z) <= 4e-8)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(x_m * y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y) * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[N[(z * z), $MachinePrecision], 4e-8], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000001e-8

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.0000000000000001e-8 < (*.f64 z z)

   1. Initial program 82.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x\_m \cdot y\right)}\\ \end{array} \end{array} \]

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 (<= y 1.2e+14)
    (/ 1.0 (* (fma (* z y) z y) x_m))
    (/ (- -1.0) (* (fma z z 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) {
	double tmp;
	if (y <= 1.2e+14) {
		tmp = 1.0 / (fma((z * y), z, y) * x_m);
	} else {
		tmp = -(-1.0) / (fma(z, z, 1.0) * (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 (y <= 1.2e+14)
		tmp = Float64(1.0 / Float64(fma(Float64(z * y), z, y) * x_m));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(z, z, 1.0) * Float64(x_m * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[y, 1.2e+14], N[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(z * z + 1.0), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e14

   1. Initial program 91.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 91.1

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

      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 91.1

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

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 94.6

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

   6. Applied rewrites 94.6%

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

   if 1.2e14 < y

   1. Initial program 91.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 96.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lower-fma.f64 96.1

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

   4. Applied rewrites 96.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(x\_m \cdot x\_m\right) \cdot y}\\ \end{array} \end{array} \]

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 z) 4e-8)
    (/ (fma (- z) z 1.0) (* x_m y))
    (/ x_m (* (* x_m 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 * z) <= 4e-8) {
		tmp = fma(-z, z, 1.0) / (x_m * y);
	} else {
		tmp = x_m / ((x_m * 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 (Float64(z * z) <= 4e-8)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(x_m * y));
	else
		tmp = Float64(x_m / Float64(Float64(x_m * x_m) * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[N[(z * z), $MachinePrecision], 4e-8], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000001e-8

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.0000000000000001e-8 < (*.f64 z z)

   1. Initial program 82.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 20.6

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

   5. Applied rewrites 20.6%

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

   7. Applied rewrites 20.3%

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

   9. Applied rewrites 27.3%

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

   11. Applied rewrites 29.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot z, x\_m, x\_m\right) \cdot y}\\ \end{array} \end{array} \]

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 (<= y 5e-40)
    (/ 1.0 (* (fma (* z y) z y) x_m))
    (/ 1.0 (* (fma (* z z) x_m 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 (y <= 5e-40) {
		tmp = 1.0 / (fma((z * y), z, y) * x_m);
	} else {
		tmp = 1.0 / (fma((z * z), x_m, 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 (y <= 5e-40)
		tmp = Float64(1.0 / Float64(fma(Float64(z * y), z, y) * x_m));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(z * z), x_m, x_m) * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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[y, 5e-40], N[(1.0 / N[(N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999965e-40

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      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 90.8

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

      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 90.8

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

   4. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 94.5

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

   6. Applied rewrites 94.5%

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

   if 4.99999999999999965e-40 < y

   1. Initial program 91.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 91.3

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

      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 91.3

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

   4. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 97.5

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 96.3

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

   9. Applied rewrites 96.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot y, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z \cdot z, x, x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\_m\right) \cdot y}\\ \end{array} \end{array} \]

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 0.85)
    (/ (fma (- z) z 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 <= 0.85) {
		tmp = fma(-z, z, 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 <= 0.85)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(x_m * y));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * x_m) * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 0.85], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.849999999999999978

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.849999999999999978 < z

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      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 85.0

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

      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 85.0

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

   4. Applied rewrites 85.0%

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

   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 84.9

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

   7. Applied rewrites 84.9%

      \[\leadsto \frac{1}{\left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x} \]
   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 86.5

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

   9. Applied rewrites 86.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(x\_m \cdot y\right)}\\ \end{array} \end{array} \]

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 0.85)
    (/ (fma (- z) z 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 <= 0.85) {
		tmp = fma(-z, z, 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 <= 0.85)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(x_m * y));
	else
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(x_m * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 0.85], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.849999999999999978

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if 0.849999999999999978 < z

   1. Initial program 84.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 84.9

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

   5. Applied rewrites 84.9%

      \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)}} \]
   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 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(x \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 91.2%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 91.0

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

   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 91.0

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

4. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 97.9

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

6. Applied rewrites 97.9%

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

7. Final simplification 97.9%

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

Alternative 9: 58.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{x\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(x\_m \cdot x\_m\right) \cdot y}\\ \end{array} \end{array} \]

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 (<= x_m 4e+31) (/ 1.0 (* x_m y)) (/ x_m (* (* x_m 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 (x_m <= 4e+31) {
		tmp = 1.0 / (x_m * y);
	} else {
		tmp = x_m / ((x_m * 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 (x_m <= 4d+31) then
        tmp = 1.0d0 / (x_m * y)
    else
        tmp = x_m / ((x_m * 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 (x_m <= 4e+31) {
		tmp = 1.0 / (x_m * y);
	} else {
		tmp = x_m / ((x_m * 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 x_m <= 4e+31:
		tmp = 1.0 / (x_m * y)
	else:
		tmp = x_m / ((x_m * 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 (x_m <= 4e+31)
		tmp = Float64(1.0 / Float64(x_m * y));
	else
		tmp = Float64(x_m / Float64(Float64(x_m * 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 (x_m <= 4e+31)
		tmp = 1.0 / (x_m * y);
	else
		tmp = x_m / ((x_m * 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[x$95$m, 4e+31], N[(1.0 / N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e31

   1. Initial program 89.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 60.3

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

   5. Applied rewrites 60.3%

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

   if 3.9999999999999999e31 < x

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.4%

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

   9. Applied rewrites 62.5%

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

   11. Applied rewrites 62.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.3× speedup

Math

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

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 (* (fma (* z z) x_m 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 / (fma((z * z), x_m, 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(fma(Float64(z * z), x_m, 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[(N[(N[(z * z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.2%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 91.0

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

   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 91.0

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

4. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 98.0

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

6. Applied rewrites 98.0%

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

7. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 90.6

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

9. Applied rewrites 90.6%

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

Alternative 11: 59.1% accurate, 2.1× 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 91.2%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 60.7

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

5. Applied rewrites 60.7%

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

6. Final simplification 60.7%

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

Developer Target 1: 93.0% 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 2024249 
(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)))))