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

Percentage Accurate: 90.7% → 98.3%

Time: 10.0s

Alternatives: 8

Speedup: 1.2×

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: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(z \cdot y\_m\right) \cdot x, z, x \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(x \cdot z, z, x\right) \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4e-23)
    (/ 1.0 (fma (* (* z y_m) x) z (* x y_m)))
    (/ (- -1.0) (* (fma (* x z) z x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4e-23) {
		tmp = 1.0 / fma(((z * y_m) * x), z, (x * y_m));
	} else {
		tmp = -(-1.0) / (fma((x * z), z, x) * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4e-23)
		tmp = Float64(1.0 / fma(Float64(Float64(z * y_m) * x), z, Float64(x * y_m)));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(Float64(x * z), z, x) * y_m));
	end
	return 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4e-23], N[(1.0 / N[(N[(N[(z * y$95$m), $MachinePrecision] * x), $MachinePrecision] * z + N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(N[(x * z), $MachinePrecision] * z + x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999984e-23

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      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 91.5

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

      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 91.5

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

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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 99.1

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

   6. Applied rewrites 99.1%

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

   if 3.99999999999999984e-23 < y

   1. Initial program 94.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 98.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 98.5

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

   6. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(z \cdot y\right) \cdot x, z, x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(x \cdot z, z, x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.9× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e+306)
    (/ 1.0 (* (* (fma z z 1.0) x) y_m))
    (/ -1.0 (* (* (- z) y_m) (* x z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+306) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y_m);
	} else {
		tmp = -1.0 / ((-z * y_m) * (x * z));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+306)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(Float64(-z) * y_m) * Float64(x * z)));
	end
	return 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+306], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[((-z) * y$95$m), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 97.3

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

      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 97.3

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

   4. Applied rewrites 97.3%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if 2.00000000000000003e306 < (*.f64 z z)

   1. Initial program 77.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 76.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 77.0

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

   6. Applied rewrites 77.0%

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

   7. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

4. Final simplification 95.6%

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

Alternative 3: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 0.001)
    (/ (fma (- z) z 1.0) (* x y_m))
    (/ 1.0 (* (* (* z z) y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = fma(-z, z, 1.0) / (x * y_m);
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(fma(Float64(-z), z, 1.0) / Float64(x * y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y_m) * x));
	end
	return 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.001], N[(N[((-z) * z + 1.0), $MachinePrecision] / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 85.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, z, 1\right)}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ -1.0 (fma (* (* (- x) z) y_m) z (* (- x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (-1.0 / fma(((-x * z) * y_m), z, (-x * y_m)));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(-1.0 / fma(Float64(Float64(Float64(-x) * z) * y_m), z, Float64(Float64(-x) * 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(-1.0 / N[(N[(N[((-x) * z), $MachinePrecision] * y$95$m), $MachinePrecision] * z + N[((-x) * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. 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. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 91.2

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

4. Applied rewrites 91.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 92.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 90.9

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

6. Applied rewrites 90.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft1-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 93.5

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

8. Applied rewrites 93.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. remove-double-neg N/A

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

   7. lift-neg.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. div-inv N/A

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

   12. lift-neg.f64 N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. clear-num N/A

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

   16. /-rgt-identity N/A

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

   17. lift-neg.f64 N/A

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

   18. metadata-eval N/A

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

   19. frac-2neg N/A

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

   20. times-frac N/A

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

   21. *-lft-identity N/A

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

   22. associate-/r/ N/A

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

   23. clear-num N/A

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

   24. lift-/.f64 N/A

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

   25. lower-fma.f64 N/A

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

10. Applied rewrites 97.1%

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

11. Final simplification 97.1%

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

Alternative 5: 94.1% accurate, 1.2× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ (- -1.0) (* (fma (* x z) z x) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (-(-1.0) / (fma((x * z), z, x) * y_m));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(Float64(-(-1.0)) / Float64(fma(Float64(x * z), z, x) * 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[((--1.0) / N[(N[(N[(x * z), $MachinePrecision] * z + x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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. 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. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 91.2

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

4. Applied rewrites 91.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 92.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 90.9

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

6. Applied rewrites 90.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft1-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 93.5

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

8. Applied rewrites 93.5%

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

9. Final simplification 93.5%

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

Alternative 6: 90.4% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ 1.0 (* (* (fma z z 1.0) x) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / ((fma(z, z, 1.0) * x) * y_m));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 92.4

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

   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 92.4

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

4. Applied rewrites 92.4%

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 90.9

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

6. Applied rewrites 90.9%

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

Alternative 7: 90.4% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (/ 1.0 (* (* (fma z z 1.0) y_m) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / ((fma(z, z, 1.0) * y_m) * x));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y_m) * x)))
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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 92.4

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

   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 92.4

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

4. Applied rewrites 92.4%

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

5. Final simplification 92.4%

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

Alternative 8: 58.0% accurate, 2.1× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (/ 1.0 (* x y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (1.0 / (x * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (1.0d0 / (x * y_m))
end function

Java

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

Python

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / Float64(x * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (1.0 / (x * 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]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(x * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 57.3

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

5. Applied rewrites 57.3%

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

6. Final simplification 57.3%

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

Developer Target 1: 92.6% 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 2024271 
(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)))))