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

Percentage Accurate: 90.9% → 98.0%

Time: 14.1s

Alternatives: 10

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% 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: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z_m \cdot z_m\right)\\ y_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{1}{x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z_m\right) \cdot \left(y_m \cdot \left(z_m \cdot x\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (if (<= t_0 5e+298)
      (/ 1.0 (* x t_0))
      (/ 1.0 (* (hypot 1.0 z_m) (* y_m (* z_m x))))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 5e+298) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = 1.0 / (hypot(1.0, z_m) * (y_m * (z_m * x)));
	}
	return y_s * tmp;
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 5e+298) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = 1.0 / (Math.hypot(1.0, z_m) * (y_m * (z_m * x)));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 5e+298:
		tmp = 1.0 / (x * t_0)
	else:
		tmp = 1.0 / (math.hypot(1.0, z_m) * (y_m * (z_m * x)))
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 5e+298)
		tmp = Float64(1.0 / Float64(x * t_0));
	else
		tmp = Float64(1.0 / Float64(hypot(1.0, z_m) * Float64(y_m * Float64(z_m * x))));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 5e+298)
		tmp = 1.0 / (x * t_0);
	else
		tmp = 1.0 / (hypot(1.0, z_m) * (y_m * (z_m * x)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e+298], N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[(y$95$m * N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5.0000000000000003e298

   1. Initial program 96.5%

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

      1. associate-/l/ 96.5%

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

   3. Simplified 96.5%

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

   if 5.0000000000000003e298 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.4%

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

      1. associate-/l/ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-*r/ 76.4%

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

      4. associate-/l/ 76.4%

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

      5. associate-*r/ 76.4%

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

      6. associate-/l* 76.4%

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

      7. associate-/r/ 76.4%

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

      8. /-rgt-identity 76.4%

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

      9. associate-*l* 83.4%

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

      10. *-commutative 83.4%

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

      11. sqr-neg 83.4%

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

      12. +-commutative 83.4%

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

      13. sqr-neg 83.4%

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

      14. fma-def 83.4%

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

   3. Simplified 83.4%

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

      1. /-rgt-identity 83.4%

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

      2. fma-udef 83.4%

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

      3. +-commutative 83.4%

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

      4. *-commutative 83.4%

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

      5. associate-*l* 76.4%

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

      6. associate-/l* 76.4%

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

      7. add-sqr-sqrt 76.4%

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

      8. associate-/l* 76.4%

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

      9. *-commutative 76.4%

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

      10. sqrt-prod 76.4%

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

      11. hypot-1-def 76.4%

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

      12. *-commutative 76.4%

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

      13. sqrt-prod 83.3%

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

      14. hypot-1-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around inf 85.6%

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

      1. expm1-log1p-u 42.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}{\frac{\frac{1}{x}}{\sqrt{y} \cdot z}}\right)\right)}} \]

      2. expm1-udef 42.2%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}{\frac{\frac{1}{x}}{\sqrt{y} \cdot z}}\right)} - 1}} \]

   9. Applied egg-rr 42.2%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{y} \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \left(\sqrt{y} \cdot \left(z \cdot x\right)\right)\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 42.2%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{y} \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \left(\sqrt{y} \cdot \left(z \cdot x\right)\right)\right)\right)\right)}} \]

       2. expm1-log1p 85.6%

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

       3. associate-*r* 85.6%

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

       4. *-commutative 85.6%

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

       5. associate-*r* 85.6%

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

       6. associate-*r* 85.6%

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

       7. rem-square-sqrt 85.7%

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

       8. *-commutative 85.7%

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

   11. Simplified 85.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z_m\right) \cdot \sqrt{y_m}\\ y_s \cdot \frac{1}{\frac{t_0}{\frac{\frac{1}{x}}{t_0}}} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z_m) (sqrt y_m))))
   (* y_s (/ 1.0 (/ t_0 (/ (/ 1.0 x) t_0))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = hypot(1.0, z_m) * sqrt(y_m);
	return y_s * (1.0 / (t_0 / ((1.0 / x) / t_0)));
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = Math.hypot(1.0, z_m) * Math.sqrt(y_m);
	return y_s * (1.0 / (t_0 / ((1.0 / x) / t_0)));
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = math.hypot(1.0, z_m) * math.sqrt(y_m)
	return y_s * (1.0 / (t_0 / ((1.0 / x) / t_0)))

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(hypot(1.0, z_m) * sqrt(y_m))
	return Float64(y_s * Float64(1.0 / Float64(t_0 / Float64(Float64(1.0 / x) / t_0))))
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z_m)
	t_0 = hypot(1.0, z_m) * sqrt(y_m);
	tmp = y_s * (1.0 / (t_0 / ((1.0 / x) / t_0)));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(1.0 / N[(t$95$0 / N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-*r/ 93.3%

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

   4. associate-/l/ 93.3%

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

   5. associate-*r/ 93.3%

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

   6. associate-/l* 93.2%

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

   7. associate-/r/ 93.3%

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

   8. /-rgt-identity 93.3%

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

   9. associate-*l* 92.6%

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

   10. *-commutative 92.6%

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

   11. sqr-neg 92.6%

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

   12. +-commutative 92.6%

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

   13. sqr-neg 92.6%

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

   14. fma-def 92.6%

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

3. Simplified 92.6%

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

   1. /-rgt-identity 92.6%

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

   2. fma-udef 92.6%

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

   3. +-commutative 92.6%

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

   4. *-commutative 92.6%

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

   5. associate-*l* 93.3%

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

   6. associate-/l* 93.2%

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

   7. add-sqr-sqrt 53.0%

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

   8. associate-/l* 53.0%

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

   9. *-commutative 53.0%

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

   10. sqrt-prod 53.0%

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

   11. hypot-1-def 53.0%

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

   12. *-commutative 53.0%

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

   13. sqrt-prod 54.1%

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

   14. hypot-1-def 56.6%

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

6. Applied egg-rr 56.6%

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

7. Final simplification 56.6%

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

Alternative 3: 49.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot {\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z_m\right) \cdot \sqrt{y_m}}\right)}^{2} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (pow (/ (pow x -0.5) (* (hypot 1.0 z_m) (sqrt y_m))) 2.0)))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	return y_s * pow((pow(x, -0.5) / (hypot(1.0, z_m) * sqrt(y_m))), 2.0);
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	return y_s * Math.pow((Math.pow(x, -0.5) / (Math.hypot(1.0, z_m) * Math.sqrt(y_m))), 2.0);
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	return y_s * math.pow((math.pow(x, -0.5) / (math.hypot(1.0, z_m) * math.sqrt(y_m))), 2.0)

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	return Float64(y_s * (Float64((x ^ -0.5) / Float64(hypot(1.0, z_m) * sqrt(y_m))) ^ 2.0))
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z_m)
	tmp = y_s * (((x ^ -0.5) / (hypot(1.0, z_m) * sqrt(y_m))) ^ 2.0);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * N[Power[N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-*r/ 93.3%

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

   4. associate-/l/ 93.3%

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

   5. associate-*r/ 93.3%

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

   6. associate-/l* 93.2%

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

   7. associate-/r/ 93.3%

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

   8. /-rgt-identity 93.3%

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

   9. associate-*l* 92.6%

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

   10. *-commutative 92.6%

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

   11. sqr-neg 92.6%

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

   12. +-commutative 92.6%

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

   13. sqr-neg 92.6%

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

   14. fma-def 92.6%

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

3. Simplified 92.6%

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

   1. fma-udef 92.6%

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

   2. +-commutative 92.6%

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

   3. *-commutative 92.6%

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

   4. associate-*l* 93.3%

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

   5. associate-/l/ 93.3%

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

   6. add-sqr-sqrt 61.4%

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

   7. sqrt-div 26.6%

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

   8. inv-pow 26.6%

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

   9. sqrt-pow1 26.6%

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

   10. metadata-eval 26.6%

       \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   11. *-commutative 26.6%

       \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   12. sqrt-prod 26.6%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   13. hypot-1-def 26.6%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)} \cdot \sqrt{y}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   14. sqrt-div 26.6%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}}} \]

   15. inv-pow 26.6%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   16. sqrt-pow1 26.5%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   17. metadata-eval 26.5%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   18. *-commutative 26.5%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \]

6. Applied egg-rr 28.7%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}} \]
7. Step-by-step derivation

   1. unpow2 28.7%

      \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

8. Simplified 28.7%

   \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

9. Final simplification 28.7%

   \[\leadsto {\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 47.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \frac{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z_m\right)}\right)}^{2}}{y_m} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (/ (pow (/ (pow x -0.5) (hypot 1.0 z_m)) 2.0) y_m)))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	return y_s * (pow((pow(x, -0.5) / hypot(1.0, z_m)), 2.0) / y_m);
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	return y_s * (Math.pow((Math.pow(x, -0.5) / Math.hypot(1.0, z_m)), 2.0) / y_m);
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	return y_s * (math.pow((math.pow(x, -0.5) / math.hypot(1.0, z_m)), 2.0) / y_m)

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	return Float64(y_s * Float64((Float64((x ^ -0.5) / hypot(1.0, z_m)) ^ 2.0) / y_m))
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z_m)
	tmp = y_s * ((((x ^ -0.5) / hypot(1.0, z_m)) ^ 2.0) / y_m);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * N[(N[Power[N[(N[Power[x, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-*r/ 93.3%

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

   4. associate-/l/ 93.3%

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

   5. associate-*r/ 93.3%

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

   6. associate-/l* 93.2%

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

   7. associate-/r/ 93.3%

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

   8. /-rgt-identity 93.3%

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

   9. associate-*l* 92.6%

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

   10. *-commutative 92.6%

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

   11. sqr-neg 92.6%

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

   12. +-commutative 92.6%

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

   13. sqr-neg 92.6%

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

   14. fma-def 92.6%

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

3. Simplified 92.6%

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

   1. /-rgt-identity 92.6%

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

   2. fma-udef 92.6%

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

   3. +-commutative 92.6%

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

   4. *-commutative 92.6%

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

   5. associate-*l* 93.3%

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

   6. associate-/l* 93.2%

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

   7. associate-/l* 92.5%

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

   8. +-commutative 92.5%

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

   9. fma-udef 92.5%

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

6. Applied egg-rr 92.5%

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

   1. clear-num 92.5%

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

   2. add-sqr-sqrt 52.8%

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

   3. *-un-lft-identity 52.8%

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

   4. times-frac 52.8%

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

   5. sqrt-div 44.4%

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

   6. inv-pow 44.4%

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

   7. sqrt-pow1 44.5%

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

   8. metadata-eval 44.5%

      \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   9. sqrt-div 46.3%

      \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{y} \]

   10. inv-pow 46.3%

       \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   11. sqrt-pow1 46.3%

       \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   12. metadata-eval 46.3%

       \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

8. Applied egg-rr 46.3%

   \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]
9. Step-by-step derivation

   1. /-rgt-identity 46.3%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   2. associate-*r/ 44.4%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]

   3. unpow2 44.4%

      \[\leadsto \frac{\color{blue}{{\left(\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right)}^{2}}}{y} \]

   4. fma-udef 44.4%

      \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{z \cdot z + 1}}}\right)}^{2}}{y} \]

   5. unpow2 44.4%

      \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{{z}^{2}} + 1}}\right)}^{2}}{y} \]

   6. +-commutative 44.4%

      \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + {z}^{2}}}}\right)}^{2}}{y} \]

   7. unpow2 44.4%

      \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{1 + \color{blue}{z \cdot z}}}\right)}^{2}}{y} \]

   8. hypot-1-def 45.5%

      \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}}\right)}^{2}}{y} \]

10. Simplified 45.5%

    \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}}{y}} \]

11. Final simplification 45.5%

    \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}}{y} \]
12. Add Preprocessing

Alternative 5: 94.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(z_m \cdot y_m, z_m, y_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z_m} \cdot \frac{\frac{1}{x}}{z_m}}{y_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 5e+65)
    (/ 1.0 (* x (fma (* z_m y_m) z_m y_m)))
    (/ (* (/ 1.0 z_m) (/ (/ 1.0 x) z_m)) y_m))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+65) {
		tmp = 1.0 / (x * fma((z_m * y_m), z_m, y_m));
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x) / z_m)) / y_m;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+65)
		tmp = Float64(1.0 / Float64(x * fma(Float64(z_m * y_m), z_m, y_m)));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x) / z_m)) / y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+65], N[(1.0 / N[(x * N[(N[(z$95$m * y$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999973e65

   1. Initial program 99.6%

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

      1. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. fma-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 4.99999999999999973e65 < (*.f64 z z)

   1. Initial program 85.2%

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

      1. associate-/l/ 85.0%

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

      2. metadata-eval 85.0%

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

      3. associate-*r/ 85.0%

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

      4. associate-/l/ 85.2%

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

      5. associate-*r/ 85.2%

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

      6. associate-/l* 84.9%

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

      7. associate-/r/ 85.0%

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

      8. /-rgt-identity 85.0%

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

      9. associate-*l* 83.4%

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

      10. *-commutative 83.4%

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

      11. sqr-neg 83.4%

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

      12. +-commutative 83.4%

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

      13. sqr-neg 83.4%

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

      14. fma-def 83.4%

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

   3. Simplified 83.4%

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

      1. /-rgt-identity 83.4%

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

      2. fma-udef 83.4%

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

      3. +-commutative 83.4%

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

      4. *-commutative 83.4%

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

      5. associate-*l* 85.0%

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

      6. associate-/l* 84.9%

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

      7. associate-/l* 83.3%

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

      8. +-commutative 83.3%

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

      9. fma-udef 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. clear-num 83.3%

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

      2. add-sqr-sqrt 61.9%

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

      3. *-un-lft-identity 61.9%

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

      4. times-frac 61.9%

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

      5. sqrt-div 42.6%

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

      6. inv-pow 42.6%

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

      7. sqrt-pow1 42.6%

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

      8. metadata-eval 42.6%

         \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      9. sqrt-div 46.9%

         \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{y} \]

      10. inv-pow 46.9%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      11. sqrt-pow1 46.9%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      12. metadata-eval 46.9%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   8. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 46.9%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      2. associate-*r/ 42.5%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]

      3. unpow2 42.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right)}^{2}}}{y} \]

      4. fma-udef 42.5%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{z \cdot z + 1}}}\right)}^{2}}{y} \]

      5. unpow2 42.5%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{{z}^{2}} + 1}}\right)}^{2}}{y} \]

      6. +-commutative 42.5%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + {z}^{2}}}}\right)}^{2}}{y} \]

      7. unpow2 42.5%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{1 + \color{blue}{z \cdot z}}}\right)}^{2}}{y} \]

      8. hypot-1-def 45.0%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}}\right)}^{2}}{y} \]

   10. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}}{y}} \]

   11. Taylor expanded in z around inf 83.3%

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

       1. associate-/r* 83.3%

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

   13. Simplified 83.3%

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

       1. *-un-lft-identity 83.3%

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

       2. unpow2 83.3%

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

       3. times-frac 87.9%

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

   15. Applied egg-rr 87.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z_m \cdot z_m\right)\\ y_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 10^{+305}:\\ \;\;\;\;\frac{1}{x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z_m}}{y_m} \cdot \frac{\frac{1}{z_m}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (if (<= t_0 1e+305)
      (/ 1.0 (* x t_0))
      (* (/ (/ 1.0 z_m) y_m) (/ (/ 1.0 z_m) x))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1d+305) then
        tmp = 1.0d0 / (x * t_0)
    else
        tmp = ((1.0d0 / z_m) / y_m) * ((1.0d0 / z_m) / x)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1e+305:
		tmp = 1.0 / (x * t_0)
	else:
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1e+305)
		tmp = Float64(1.0 / Float64(x * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y_m) * Float64(Float64(1.0 / z_m) / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1e+305)
		tmp = 1.0 / (x * t_0);
	else
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 1e+305], N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 9.9999999999999994e304

   1. Initial program 96.5%

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

      1. associate-/l/ 96.5%

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

   3. Simplified 96.5%

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

   if 9.9999999999999994e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.4%

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

      1. associate-/l/ 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-*r/ 76.4%

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

      4. associate-/l/ 76.4%

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

      5. associate-*r/ 76.4%

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

      6. associate-/l* 76.4%

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

      7. associate-/r/ 76.4%

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

      8. /-rgt-identity 76.4%

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

      9. associate-*l* 83.4%

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

      10. *-commutative 83.4%

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

      11. sqr-neg 83.4%

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

      12. +-commutative 83.4%

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

      13. sqr-neg 83.4%

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

      14. fma-def 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in z around inf 76.4%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

   7. Simplified 83.1%

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

      1. expm1-log1p-u 83.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 76.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 76.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 76.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 76.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 84.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 84.0%

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

   11. Simplified 84.0%

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

       1. add-sqr-sqrt 83.9%

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

       2. *-commutative 83.9%

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

       3. times-frac 81.9%

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

       4. sqrt-pow1 78.7%

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

       5. metadata-eval 78.7%

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

       6. unpow-1 78.7%

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

       7. sqrt-pow1 95.2%

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

       8. metadata-eval 95.2%

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

       9. unpow-1 95.2%

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

   13. Applied egg-rr 95.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y} \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 1.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x \cdot \left(y_m \cdot \left(1 + z_m \cdot z_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z_m} \cdot \frac{\frac{1}{x}}{z_m}}{y_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 1.5e+69)
    (/ 1.0 (* x (* y_m (+ 1.0 (* z_m z_m)))))
    (/ (* (/ 1.0 z_m) (/ (/ 1.0 x) z_m)) y_m))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1.5e+69) {
		tmp = 1.0 / (x * (y_m * (1.0 + (z_m * z_m))));
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x) / z_m)) / y_m;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1.5d+69) then
        tmp = 1.0d0 / (x * (y_m * (1.0d0 + (z_m * z_m))))
    else
        tmp = ((1.0d0 / z_m) * ((1.0d0 / x) / z_m)) / y_m
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1.5e+69) {
		tmp = 1.0 / (x * (y_m * (1.0 + (z_m * z_m))));
	} else {
		tmp = ((1.0 / z_m) * ((1.0 / x) / z_m)) / y_m;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if (z_m * z_m) <= 1.5e+69:
		tmp = 1.0 / (x * (y_m * (1.0 + (z_m * z_m))))
	else:
		tmp = ((1.0 / z_m) * ((1.0 / x) / z_m)) / y_m
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1.5e+69)
		tmp = Float64(1.0 / Float64(x * Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x) / z_m)) / y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1.5e+69)
		tmp = 1.0 / (x * (y_m * (1.0 + (z_m * z_m))));
	else
		tmp = ((1.0 / z_m) * ((1.0 / x) / z_m)) / y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1.5e+69], N[(1.0 / N[(x * N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.49999999999999992e69

   1. Initial program 99.6%

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

      1. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   if 1.49999999999999992e69 < (*.f64 z z)

   1. Initial program 85.1%

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

      1. associate-/l/ 84.9%

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

      2. metadata-eval 84.9%

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

      3. associate-*r/ 84.9%

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

      4. associate-/l/ 85.1%

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

      5. associate-*r/ 85.1%

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

      6. associate-/l* 84.8%

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

      7. associate-/r/ 84.9%

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

      8. /-rgt-identity 84.9%

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

      9. associate-*l* 83.2%

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

      10. *-commutative 83.2%

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

      11. sqr-neg 83.2%

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

      12. +-commutative 83.2%

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

      13. sqr-neg 83.2%

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

      14. fma-def 83.2%

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

   3. Simplified 83.2%

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

      1. /-rgt-identity 83.2%

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

      2. fma-udef 83.2%

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

      3. +-commutative 83.2%

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

      4. *-commutative 83.2%

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

      5. associate-*l* 84.9%

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

      6. associate-/l* 84.8%

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

      7. associate-/l* 83.1%

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

      8. +-commutative 83.1%

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

      9. fma-udef 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. clear-num 83.2%

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

      2. add-sqr-sqrt 62.4%

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

      3. *-un-lft-identity 62.4%

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

      4. times-frac 62.4%

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

      5. sqrt-div 43.0%

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

      6. inv-pow 43.0%

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

      7. sqrt-pow1 43.0%

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

      8. metadata-eval 43.0%

         \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      9. sqrt-div 47.4%

         \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{y} \]

      10. inv-pow 47.4%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      11. sqrt-pow1 47.3%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      12. metadata-eval 47.3%

          \[\leadsto \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

   8. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{1} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]
   9. Step-by-step derivation

      1. /-rgt-identity 47.3%

         \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y} \]

      2. associate-*r/ 42.9%

         \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}} \]

      3. unpow2 42.9%

         \[\leadsto \frac{\color{blue}{{\left(\frac{{x}^{-0.5}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right)}^{2}}}{y} \]

      4. fma-udef 42.9%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{z \cdot z + 1}}}\right)}^{2}}{y} \]

      5. unpow2 42.9%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{{z}^{2}} + 1}}\right)}^{2}}{y} \]

      6. +-commutative 42.9%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + {z}^{2}}}}\right)}^{2}}{y} \]

      7. unpow2 42.9%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\sqrt{1 + \color{blue}{z \cdot z}}}\right)}^{2}}{y} \]

      8. hypot-1-def 45.5%

         \[\leadsto \frac{{\left(\frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}}\right)}^{2}}{y} \]

   10. Simplified 45.5%

       \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}}{y}} \]

   11. Taylor expanded in z around inf 83.2%

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

       1. associate-/r* 83.2%

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

   13. Simplified 83.2%

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

       1. *-un-lft-identity 83.2%

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

       2. unpow2 83.2%

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

       3. times-frac 87.8%

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

   15. Applied egg-rr 87.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 0.52:\\ \;\;\;\;\frac{1}{y_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z_m}}{y_m} \cdot \frac{\frac{1}{z_m}}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= z_m 0.52)
    (/ 1.0 (* y_m x))
    (* (/ (/ 1.0 z_m) y_m) (/ (/ 1.0 z_m) x)))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.52) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 0.52d0) then
        tmp = 1.0d0 / (y_m * x)
    else
        tmp = ((1.0d0 / z_m) / y_m) * ((1.0d0 / z_m) / x)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.52) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 0.52:
		tmp = 1.0 / (y_m * x)
	else:
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x)
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.52)
		tmp = Float64(1.0 / Float64(y_m * x));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y_m) * Float64(Float64(1.0 / z_m) / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.52)
		tmp = 1.0 / (y_m * x);
	else
		tmp = ((1.0 / z_m) / y_m) * ((1.0 / z_m) / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 0.52], N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.52000000000000002

   1. Initial program 94.8%

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

      1. associate-/l/ 94.9%

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

      2. metadata-eval 94.9%

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

      3. associate-*r/ 94.9%

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

      4. associate-/l/ 94.8%

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

      5. associate-*r/ 94.8%

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

      6. associate-/l* 94.8%

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

      7. associate-/r/ 94.9%

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

      8. /-rgt-identity 94.9%

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

      9. associate-*l* 93.5%

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

      10. *-commutative 93.5%

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

      11. sqr-neg 93.5%

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

      12. +-commutative 93.5%

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

      13. sqr-neg 93.5%

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

      14. fma-def 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in z around 0 73.1%

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

   if 0.52000000000000002 < z

   1. Initial program 88.5%

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

      1. associate-/l/ 88.1%

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

      2. metadata-eval 88.1%

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

      3. associate-*r/ 88.1%

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

      4. associate-/l/ 88.5%

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

      5. associate-*r/ 88.5%

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

      6. associate-/l* 88.1%

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

      7. associate-/r/ 88.1%

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

      8. /-rgt-identity 88.1%

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

      9. associate-*l* 89.7%

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

      10. *-commutative 89.7%

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

      11. sqr-neg 89.7%

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

      12. +-commutative 89.7%

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

      13. sqr-neg 89.7%

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

      14. fma-def 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in z around inf 85.8%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. expm1-log1p-u 78.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 51.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 80.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 87.6%

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

   11. Simplified 87.6%

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

       1. add-sqr-sqrt 87.5%

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

       2. *-commutative 87.5%

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

       3. times-frac 87.2%

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

       4. sqrt-pow1 87.3%

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

       5. metadata-eval 87.3%

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

       6. unpow-1 87.3%

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

       7. sqrt-pow1 90.6%

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

       8. metadata-eval 90.6%

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

       9. unpow-1 90.6%

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

   13. Applied egg-rr 90.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.52:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y} \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 0.52:\\ \;\;\;\;\frac{1}{y_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z_m \cdot y_m\right)}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= z_m 0.52) (/ 1.0 (* y_m x)) (/ 1.0 (* x (* z_m y_m))))))

C

z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.52) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = 1.0 / (x * (z_m * y_m));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 0.52d0) then
        tmp = 1.0d0 / (y_m * x)
    else
        tmp = 1.0d0 / (x * (z_m * y_m))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 0.52) {
		tmp = 1.0 / (y_m * x);
	} else {
		tmp = 1.0 / (x * (z_m * y_m));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 0.52:
		tmp = 1.0 / (y_m * x)
	else:
		tmp = 1.0 / (x * (z_m * y_m))
	return y_s * tmp

Julia

z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 0.52)
		tmp = Float64(1.0 / Float64(y_m * x));
	else
		tmp = Float64(1.0 / Float64(x * Float64(z_m * y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 0.52)
		tmp = 1.0 / (y_m * x);
	else
		tmp = 1.0 / (x * (z_m * y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 0.52], N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.52000000000000002

   1. Initial program 94.8%

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

      1. associate-/l/ 94.9%

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

      2. metadata-eval 94.9%

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

      3. associate-*r/ 94.9%

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

      4. associate-/l/ 94.8%

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

      5. associate-*r/ 94.8%

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

      6. associate-/l* 94.8%

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

      7. associate-/r/ 94.9%

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

      8. /-rgt-identity 94.9%

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

      9. associate-*l* 93.5%

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

      10. *-commutative 93.5%

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

      11. sqr-neg 93.5%

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

      12. +-commutative 93.5%

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

      13. sqr-neg 93.5%

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

      14. fma-def 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in z around 0 73.1%

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

   if 0.52000000000000002 < z

   1. Initial program 88.5%

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

      1. associate-/l/ 88.1%

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

      2. metadata-eval 88.1%

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

      3. associate-*r/ 88.1%

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

      4. associate-/l/ 88.5%

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

      5. associate-*r/ 88.5%

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

      6. associate-/l* 88.1%

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

      7. associate-/r/ 88.1%

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

      8. /-rgt-identity 88.1%

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

      9. associate-*l* 89.7%

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

      10. *-commutative 89.7%

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

      11. sqr-neg 89.7%

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

      12. +-commutative 89.7%

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

      13. sqr-neg 89.7%

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

      14. fma-def 89.7%

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

   3. Simplified 89.7%

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

      1. /-rgt-identity 89.7%

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

      2. fma-udef 89.7%

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

      3. +-commutative 89.7%

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

      4. *-commutative 89.7%

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

      5. associate-*l* 88.1%

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

      6. associate-/l* 88.1%

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

      7. add-sqr-sqrt 53.3%

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

      8. associate-/l* 53.2%

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

      9. *-commutative 53.2%

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

      10. sqrt-prod 53.3%

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

      11. hypot-1-def 53.2%

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

      12. *-commutative 53.2%

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

      13. sqrt-prod 56.4%

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

      14. hypot-1-def 59.5%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in z around inf 58.2%

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

   8. Taylor expanded in z around 0 37.9%

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

      1. *-commutative 37.9%

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

   10. Simplified 37.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.52:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z_m) :precision binary64 (* y_s (/ 1.0 (* y_m x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z_m)
	tmp = y_s * (1.0 / (y_m * x));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

   1. associate-/l/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-*r/ 93.3%

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

   4. associate-/l/ 93.3%

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

   5. associate-*r/ 93.3%

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

   6. associate-/l* 93.2%

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

   7. associate-/r/ 93.3%

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

   8. /-rgt-identity 93.3%

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

   9. associate-*l* 92.6%

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

   10. *-commutative 92.6%

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

   11. sqr-neg 92.6%

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

   12. +-commutative 92.6%

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

   13. sqr-neg 92.6%

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

   14. fma-def 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in z around 0 59.5%

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

6. Final simplification 59.5%

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

Developer target: 93.0% accurate, 0.3× 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 2024021 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

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