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

Percentage Accurate: 89.6% → 98.4%

Time: 8.8s

Alternatives: 12

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: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, x \cdot z, y\_m \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y\_m \cdot x\right) \cdot z, z, y\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.7e+129)
    (/ 1.0 (fma (* y_m z) (* x z) (* y_m x)))
    (/ 1.0 (fma (* (* y_m x) z) z (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.7e+129) {
		tmp = 1.0 / fma((y_m * z), (x * z), (y_m * x));
	} else {
		tmp = 1.0 / fma(((y_m * x) * z), z, (y_m * x));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.7e+129)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(x * z), Float64(y_m * x)));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(y_m * x) * z), z, Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.7e+129], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(x * z), $MachinePrecision] + N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * x), $MachinePrecision] * z), $MachinePrecision] * z + N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000009e129

   1. Initial program 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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 97.0

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

   6. Applied rewrites 97.0%

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

   if 1.70000000000000009e129 < y

   1. Initial program 97.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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 96.0

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

   6. Applied rewrites 96.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 98.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, x \cdot z, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y \cdot x\right) \cdot z, z, y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y\_m \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\_m\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* (+ (* z z) 1.0) y_m) 2e+304)
    (/ 1.0 (* (* (fma z z 1.0) y_m) x))
    (/ 1.0 (* (* (* z z) x) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((((z * z) + 1.0) * y_m) <= 2e+304) {
		tmp = 1.0 / ((fma(z, z, 1.0) * y_m) * x);
	} else {
		tmp = 1.0 / (((z * z) * x) * y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(z * z) + 1.0) * y_m) <= 2e+304)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * y_m) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], 2e+304], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1.9999999999999999e304

   1. Initial program 92.0%

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

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

      5. lower-*.f64 91.6

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

      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 91.6

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

   4. Applied rewrites 91.6%

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

   if 1.9999999999999999e304 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 81.9%

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

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

   5. Applied rewrites 81.9%

      \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)}} \]
   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 85.1

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

   7. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* 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 85.3

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

   9. Applied rewrites 85.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z + 1\right) \cdot y \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y\_m \cdot z\right) \cdot x, z, y\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e+300)
    (/ 1.0 (* (* (fma z z 1.0) x) y_m))
    (/ 1.0 (fma (* (* y_m z) x) z (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+300) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y_m);
	} else {
		tmp = 1.0 / fma(((y_m * z) * x), z, (y_m * x));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+300)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y_m));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(y_m * z) * x), z, Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+300], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * x), $MachinePrecision] * z + N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e300

   1. Initial program 96.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 95.9

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

      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 95.9

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

   4. Applied rewrites 95.9%

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

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. /-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-2neg N/A

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

      10. lift-neg.f64 N/A

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

      11. metadata-eval N/A

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

      12. associate-/r/ N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 96.8%

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

   if 2.0000000000000001e300 < (*.f64 z z)

   1. Initial program 74.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 74.5

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

      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 74.5

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

   4. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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 97.0

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

   6. Applied rewrites 97.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(y \cdot z\right) \cdot x, z, y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, x \cdot z, y\_m \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 7e+95)
    (/ 1.0 (fma (* y_m z) (* x z) (* y_m x)))
    (/ (- -1.0) (* (fma z z 1.0) (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 7e+95) {
		tmp = 1.0 / fma((y_m * z), (x * z), (y_m * x));
	} else {
		tmp = -(-1.0) / (fma(z, z, 1.0) * (y_m * x));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 7e+95)
		tmp = Float64(1.0 / fma(Float64(y_m * z), Float64(x * z), Float64(y_m * x)));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(z, z, 1.0) * Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 7e+95], N[(1.0 / N[(N[(y$95$m * z), $MachinePrecision] * N[(x * z), $MachinePrecision] + N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(z * z + 1.0), $MachinePrecision] * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999999e95

   1. Initial program 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 89.6

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

   4. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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 97.0

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

   6. Applied rewrites 97.0%

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

   if 6.99999999999999999e95 < y

   1. Initial program 97.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 98.5

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

      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 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, x \cdot z, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right) \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e+300)
    (/ 1.0 (* (* (fma z z 1.0) x) y_m))
    (/ 1.0 (* (fma (* y_m z) z y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+300) {
		tmp = 1.0 / ((fma(z, z, 1.0) * x) * y_m);
	} else {
		tmp = 1.0 / (fma((y_m * z), z, y_m) * x);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+300)
		tmp = Float64(1.0 / Float64(Float64(fma(z, z, 1.0) * x) * y_m));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * 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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+300], N[(1.0 / N[(N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e300

   1. Initial program 96.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 95.9

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

      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 95.9

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

   4. Applied rewrites 95.9%

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

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. /-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. frac-2neg N/A

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

      10. lift-neg.f64 N/A

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

      11. metadata-eval N/A

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

      12. associate-/r/ N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 96.8%

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

   if 2.0000000000000001e300 < (*.f64 z z)

   1. Initial program 74.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 74.5

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

      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 74.5

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

   4. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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. 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 92.4

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

   6. Applied rewrites 92.4%

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

Alternative 6: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* z z) 2e-15)
    (/ (/ -1.0 x) (- y_m))
    (/ 1.0 (* (* (* z z) y_m) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-15) {
		tmp = (-1.0 / x) / -y_m;
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if ((z * z) <= 2d-15) then
        tmp = ((-1.0d0) / x) / -y_m
    else
        tmp = 1.0d0 / (((z * z) * y_m) * x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z;
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-15) {
		tmp = (-1.0 / x) / -y_m;
	} else {
		tmp = 1.0 / (((z * z) * y_m) * x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if (z * z) <= 2e-15:
		tmp = (-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)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-15)
		tmp = Float64(Float64(-1.0 / x) / Float64(-y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z * z) * y_m) * x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e-15)
		tmp = (-1.0 / x) / -y_m;
	else
		tmp = 1.0 / (((z * z) * y_m) * x);
	end
	tmp_2 = 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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-15], N[(N[(-1.0 / x), $MachinePrecision] / (-y$95$m)), $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) < 2.0000000000000002e-15

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 2.0000000000000002e-15 < (*.f64 z z)

   1. Initial program 83.0%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{-1}{x}}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z] = \mathsf{sort}([x, y_m, z])\\ \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-42}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1e-42)
    (/ 1.0 (* (fma (* y_m z) z y_m) x))
    (/ (- -1.0) (* (fma z z 1.0) (* y_m x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-42) {
		tmp = 1.0 / (fma((y_m * z), z, y_m) * x);
	} else {
		tmp = -(-1.0) / (fma(z, z, 1.0) * (y_m * x));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1e-42)
		tmp = Float64(1.0 / Float64(fma(Float64(y_m * z), z, y_m) * x));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(z, z, 1.0) * Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1e-42], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(N[(z * z + 1.0), $MachinePrecision] * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000004e-42

   1. Initial program 89.5%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 89.3

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

      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 89.3

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

   4. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
   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-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 95.1

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

   6. Applied rewrites 95.1%

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

   if 1.00000000000000004e-42 < y

   1. Initial program 95.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-42}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot z, z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (-1.0 / x) / -y_m;
	else
		tmp = 1.0 / (((z * z) * x) * y_m);
	end
	tmp_2 = 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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.0], N[(N[(-1.0 / x), $MachinePrecision] / (-y$95$m)), $MachinePrecision], N[(1.0 / N[(N[(N[(z * z), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if 1 < z

   1. Initial program 86.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.6

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

   5. Applied rewrites 85.6%

      \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)}} \]
   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 82.0

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

   7. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* 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 85.0

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

   9. Applied rewrites 85.0%

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

Alternative 9: 75.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z] = sort([x, y_m, z])
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 1.0:
		tmp = (-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)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(-1.0 / x) / Float64(-y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(y_m * x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (-1.0 / x) / -y_m;
	else
		tmp = 1.0 / ((z * z) * (y_m * x));
	end
	tmp_2 = 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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 1.0], N[(N[(-1.0 / x), $MachinePrecision] / (-y$95$m)), $MachinePrecision], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.4%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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. 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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if 1 < z

   1. Initial program 86.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 85.6

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

   5. Applied rewrites 85.6%

      \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)}} \]
   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 82.0

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

   7. Applied rewrites 82.0%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z = sort([x, y_m, z])
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(1.0 / fma(Float64(y_m * Float64(x * z)), z, Float64(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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(N[(y$95$m * N[(x * z), $MachinePrecision]), $MachinePrecision] * z + N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 90.6

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

   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 90.6

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

4. Applied rewrites 90.6%

   \[\leadsto \color{blue}{\frac{1}{\left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right) \cdot x}} \]
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 96.9

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

6. Applied rewrites 96.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 94.5

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 94.5

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 94.5

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

8. Applied rewrites 94.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 97.9

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

10. Applied rewrites 97.9%

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

11. Final simplification 97.9%

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

Alternative 11: 91.1% accurate, 1.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
(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);
assert(x < y_m && y_m < z);
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)
x, y_m, z = sort([x, y_m, z])
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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
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 90.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 90.6

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

   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 90.6

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

4. Applied rewrites 90.6%

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

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. /-rgt-identity N/A

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

   7. associate-*r/ N/A

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

   8. associate-*l/ N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f64 N/A

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

   11. metadata-eval N/A

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

   12. associate-/r/ N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l/ N/A

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

   15. lift-/.f64 N/A

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

   16. associate-/r/ N/A

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

   17. lower-*.f64 N/A

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

6. Applied rewrites 91.3%

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

Alternative 12: 59.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z = num2cell(sort([x, y_m, z])){:}
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (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]
NOTE: x, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
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 58.0

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

5. Applied rewrites 58.0%

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

Developer Target 1: 92.5% 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 2024255 
(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)))))