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

Percentage Accurate: 90.5% → 99.1%

Time: 12.9s

Alternatives: 11

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: 90.5% 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: 99.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\_m\right) \cdot \sqrt{y\_m}} \cdot \frac{\sqrt{\frac{1}{y\_m}}}{z\_m \cdot x}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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+304)
      (/ (/ 1.0 x) t_0)
      (*
       (/ 1.0 (* (hypot 1.0 z_m) (sqrt y_m)))
       (/ (sqrt (/ 1.0 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 <= 1e+304) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (hypot(1.0, z_m) * sqrt(y_m))) * (sqrt((1.0 / 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 <= 1e+304) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (Math.hypot(1.0, z_m) * Math.sqrt(y_m))) * (Math.sqrt((1.0 / 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 <= 1e+304:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / (math.hypot(1.0, z_m) * math.sqrt(y_m))) * (math.sqrt((1.0 / 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 <= 1e+304)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, z_m) * sqrt(y_m))) * Float64(sqrt(Float64(1.0 / 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 <= 1e+304)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / (hypot(1.0, z_m) * sqrt(y_m))) * (sqrt((1.0 / 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, 1e+304], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / y$95$m), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.9999999999999994e303

   1. Initial program 94.2%

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

   if 9.9999999999999994e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 79.0%

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

      1. associate-/l/ 79.0%

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

      2. associate-*l* 82.0%

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

      3. *-commutative 82.0%

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

      4. sqr-neg 82.0%

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

      5. +-commutative 82.0%

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

      6. sqr-neg 82.0%

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

      7. fma-define 82.0%

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

   3. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*r* 79.0%

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

      3. fma-undefine 79.0%

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

      4. +-commutative 79.0%

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

      5. associate-/l/ 79.0%

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

      6. add-sqr-sqrt 79.0%

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

      7. *-un-lft-identity 79.0%

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

      8. times-frac 79.0%

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

      9. +-commutative 79.0%

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

      10. fma-undefine 79.0%

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

      11. *-commutative 79.0%

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

      12. sqrt-prod 79.0%

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

      13. fma-undefine 79.0%

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

      14. +-commutative 79.0%

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

      15. hypot-1-def 79.0%

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

      16. +-commutative 79.0%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around inf 87.8%

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

      1. associate-*l/ 87.8%

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

      2. *-lft-identity 87.8%

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

   9. Simplified 87.8%

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

4. Final simplification 93.5%

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

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\_m\right) \cdot \sqrt{y\_m}\\ y\_s \cdot \left(\frac{1}{t\_0} \cdot \frac{\frac{1}{x}}{t\_0}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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(Float64(1.0 / 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[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.4%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

   1. *-commutative 92.4%

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

   2. associate-*r* 92.0%

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

   3. fma-undefine 92.0%

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

   4. +-commutative 92.0%

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

   5. associate-/l/ 92.4%

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

   6. add-sqr-sqrt 45.3%

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

   7. *-un-lft-identity 45.3%

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

   8. times-frac 45.2%

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

   9. +-commutative 45.2%

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

   10. fma-undefine 45.2%

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

   11. *-commutative 45.2%

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

   12. sqrt-prod 45.2%

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

   13. fma-undefine 45.2%

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

   14. +-commutative 45.2%

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

   15. hypot-1-def 45.3%

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

   16. +-commutative 45.3%

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

6. Applied egg-rr 47.7%

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

Alternative 3: 49.3% 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 #s(literal 1 binary64) 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 92.4%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

   1. *-commutative 92.4%

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

   2. associate-*r* 92.0%

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

   3. fma-undefine 92.0%

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

   4. +-commutative 92.0%

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

   5. associate-/l/ 92.4%

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

   6. add-sqr-sqrt 59.2%

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

   7. sqrt-div 21.0%

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

   8. inv-pow 21.0%

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

   9. sqrt-pow1 21.0%

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

   10. metadata-eval 21.0%

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

   11. +-commutative 21.0%

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

   12. fma-undefine 21.0%

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

   13. *-commutative 21.0%

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

   14. sqrt-prod 21.0%

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

   15. fma-undefine 21.0%

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

   16. +-commutative 21.0%

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

   17. hypot-1-def 21.0%

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

   18. sqrt-div 21.0%

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

6. Applied egg-rr 21.7%

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

   1. unpow2 21.7%

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

8. Simplified 21.7%

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

Alternative 4: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z\_m, z\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z\_m \cdot y\_m}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 2e+289)
    (* (/ 1.0 y_m) (/ (/ 1.0 x) (fma z_m z_m 1.0)))
    (/ (/ (/ 1.0 x) (* z_m y_m)) z_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) <= 2e+289) {
		tmp = (1.0 / y_m) * ((1.0 / x) / fma(z_m, z_m, 1.0));
	} else {
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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) <= 2e+289)
		tmp = Float64(Float64(1.0 / y_m) * Float64(Float64(1.0 / x) / fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z_m * y_m)) / z_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], 2e+289], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e289

   1. Initial program 97.7%

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

      1. associate-/l/ 97.2%

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

      2. associate-*l* 97.7%

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

      3. *-commutative 97.7%

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

      4. sqr-neg 97.7%

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

      5. +-commutative 97.7%

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

      6. sqr-neg 97.7%

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

      7. fma-define 97.7%

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

   3. Simplified 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r* 97.2%

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

      3. fma-undefine 97.2%

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

      4. +-commutative 97.2%

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

      5. associate-/l/ 97.7%

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

      6. *-un-lft-identity 97.7%

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

      7. +-commutative 97.7%

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

      8. fma-undefine 97.7%

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

      9. times-frac 98.0%

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

   6. Applied egg-rr 98.0%

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

   if 2.0000000000000001e289 < (*.f64 z z)

   1. Initial program 78.7%

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

      1. associate-/l/ 78.7%

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

      2. associate-*l* 78.6%

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

      3. *-commutative 78.6%

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

      4. sqr-neg 78.6%

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

      5. +-commutative 78.6%

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

      6. sqr-neg 78.6%

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

      7. fma-define 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in z around inf 78.7%

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

      1. associate-/r* 78.7%

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

      2. associate-/r* 78.3%

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

   7. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. unpow2 78.3%

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

      3. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. associate-*r/ 91.7%

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

       2. frac-times 99.9%

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

       3. *-un-lft-identity 99.9%

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

   11. Applied egg-rr 99.9%

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

Alternative 5: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z\_m \cdot y\_m}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 5e+281)
    (/ 1.0 (* y_m (* x (fma z_m z_m 1.0))))
    (/ (/ (/ 1.0 x) (* z_m y_m)) z_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+281) {
		tmp = 1.0 / (y_m * (x * fma(z_m, z_m, 1.0)));
	} else {
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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+281)
		tmp = Float64(1.0 / Float64(y_m * Float64(x * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z_m * y_m)) / z_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+281], N[(1.0 / N[(y$95$m * N[(x * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000016e281

   1. Initial program 98.1%

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

      1. associate-/l/ 97.6%

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

      2. associate-*l* 97.7%

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

      3. *-commutative 97.7%

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

      4. sqr-neg 97.7%

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

      5. +-commutative 97.7%

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

      6. sqr-neg 97.7%

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

      7. fma-define 97.7%

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

   3. Simplified 97.7%

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

   if 5.00000000000000016e281 < (*.f64 z z)

   1. Initial program 78.0%

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

      1. associate-/l/ 78.0%

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

      2. associate-*l* 79.2%

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

      3. *-commutative 79.2%

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

      4. sqr-neg 79.2%

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

      5. +-commutative 79.2%

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

      6. sqr-neg 79.2%

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

      7. fma-define 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in z around inf 78.0%

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

      1. associate-/r* 78.0%

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

      2. associate-/r* 78.9%

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

   7. Simplified 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. unpow2 78.9%

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

      3. times-frac 91.9%

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

   9. Applied egg-rr 91.9%

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

       1. associate-*r/ 91.9%

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

       2. frac-times 99.9%

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

       3. *-un-lft-identity 99.9%

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

   11. Applied egg-rr 99.9%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z\_m \cdot y\_m}}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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+304) (/ (/ 1.0 x) t_0) (/ (/ (/ 1.0 x) (* z_m y_m)) z_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 t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1d+304) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = ((1.0d0 / x) / (z_m * y_m)) / z_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 t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1e+304:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1e+304)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z_m * y_m)) / z_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)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1e+304)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / x) / (z_m * y_m)) / z_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_] := 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+304], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.9999999999999994e303

   1. Initial program 94.2%

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

   if 9.9999999999999994e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 79.0%

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

      1. associate-/l/ 79.0%

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

      2. associate-*l* 82.0%

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

      3. *-commutative 82.0%

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

      4. sqr-neg 82.0%

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

      5. +-commutative 82.0%

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

      6. sqr-neg 82.0%

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

      7. fma-define 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in z around inf 79.0%

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

      1. associate-/r* 79.0%

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

      2. associate-/r* 81.8%

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

   7. Simplified 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. unpow2 81.8%

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

      3. times-frac 96.7%

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

   9. Applied egg-rr 96.7%

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

       1. associate-*r/ 96.7%

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

       2. frac-times 99.8%

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

       3. *-un-lft-identity 99.8%

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

   11. Applied egg-rr 99.8%

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

Alternative 7: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z\_m} \cdot \frac{1}{z\_m \cdot y\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 2e-11)
    (/ (/ 1.0 y_m) x)
    (* (/ (/ 1.0 x) z_m) (/ 1.0 (* 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) <= 2e-11) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = ((1.0 / x) / z_m) * (1.0 / (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) <= 2d-11) then
        tmp = (1.0d0 / y_m) / x
    else
        tmp = ((1.0d0 / x) / z_m) * (1.0d0 / (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) <= 2e-11) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = ((1.0 / x) / z_m) * (1.0 / (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) <= 2e-11:
		tmp = (1.0 / y_m) / x
	else:
		tmp = ((1.0 / x) / z_m) * (1.0 / (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) <= 2e-11)
		tmp = Float64(Float64(1.0 / y_m) / x);
	else
		tmp = Float64(Float64(Float64(1.0 / x) / z_m) * Float64(1.0 / 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 * z_m) <= 2e-11)
		tmp = (1.0 / y_m) / x;
	else
		tmp = ((1.0 / x) / z_m) * (1.0 / (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], 2e-11], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.7%

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

      1. associate-/l/ 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sqr-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. sqr-neg 99.6%

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

      7. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. fma-undefine 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-/l/ 99.7%

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

      6. add-sqr-sqrt 47.8%

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

      7. *-un-lft-identity 47.8%

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

      8. times-frac 47.7%

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

      9. +-commutative 47.7%

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

      10. fma-undefine 47.7%

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

      11. *-commutative 47.7%

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

      12. sqrt-prod 47.7%

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

      13. fma-undefine 47.7%

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

      14. +-commutative 47.7%

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

      15. hypot-1-def 47.7%

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

      16. +-commutative 47.7%

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

   6. Applied egg-rr 47.7%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

      3. clear-num 23.1%

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

      4. sqrt-div 20.7%

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

      5. metadata-eval 20.7%

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

      6. div-inv 20.7%

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

      7. clear-num 20.7%

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

      8. /-rgt-identity 20.7%

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

      9. associate-*l* 20.7%

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

   8. Applied egg-rr 20.7%

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

   9. Taylor expanded in z around 0 99.4%

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

       1. associate-/l/ 99.5%

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

   11. Simplified 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 84.9%

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

      1. associate-/l/ 84.4%

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

      2. associate-*l* 85.1%

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

      3. *-commutative 85.1%

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

      4. sqr-neg 85.1%

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

      5. +-commutative 85.1%

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

      6. sqr-neg 85.1%

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

      7. fma-define 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in z around inf 82.0%

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

      1. associate-/r* 82.5%

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

      2. associate-/r* 82.3%

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

   7. Simplified 82.3%

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

      1. *-un-lft-identity 82.3%

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

      2. unpow2 82.3%

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

      3. times-frac 89.7%

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

   9. Applied egg-rr 89.7%

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

       1. *-commutative 89.7%

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

       2. clear-num 89.7%

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

       3. frac-times 89.3%

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

       4. metadata-eval 89.3%

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

       5. associate-/l/ 89.2%

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

       6. associate-/r/ 89.5%

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

       7. /-rgt-identity 89.5%

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

   11. Applied egg-rr 89.5%

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

       1. frac-2neg 89.5%

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

       2. metadata-eval 89.5%

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

       3. *-commutative 89.5%

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

       4. distribute-lft-neg-in 89.5%

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

       5. associate-*r* 95.1%

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

       6. associate-*l* 88.7%

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

       7. associate-/l/ 89.2%

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

       8. associate-/r* 95.6%

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

       9. div-inv 95.5%

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

       10. frac-2neg 95.5%

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

       11. add-sqr-sqrt 48.4%

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

       12. sqrt-unprod 59.7%

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

       13. sqr-neg 59.7%

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

       14. sqrt-unprod 25.2%

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

       15. add-sqr-sqrt 50.5%

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

       16. metadata-eval 50.5%

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

       17. add-sqr-sqrt 25.9%

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

       18. sqrt-unprod 68.7%

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

       19. sqr-neg 68.7%

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

       20. sqrt-prod 48.7%

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

       21. add-sqr-sqrt 95.5%

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

   13. Applied egg-rr 95.5%

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

Alternative 8: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z\_m \cdot \left(x \cdot \left(z\_m \cdot y\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (* z_m z_m) 2e-11)
    (/ (/ 1.0 y_m) x)
    (/ 1.0 (* z_m (* 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) <= 2e-11) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / (z_m * (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) <= 2d-11) then
        tmp = (1.0d0 / y_m) / x
    else
        tmp = 1.0d0 / (z_m * (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) <= 2e-11) {
		tmp = (1.0 / y_m) / x;
	} else {
		tmp = 1.0 / (z_m * (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) <= 2e-11:
		tmp = (1.0 / y_m) / x
	else:
		tmp = 1.0 / (z_m * (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) <= 2e-11)
		tmp = Float64(Float64(1.0 / y_m) / x);
	else
		tmp = Float64(1.0 / Float64(z_m * 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 * z_m) <= 2e-11)
		tmp = (1.0 / y_m) / x;
	else
		tmp = 1.0 / (z_m * (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], 2e-11], N[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(z$95$m * N[(x * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.7%

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

      1. associate-/l/ 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sqr-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. sqr-neg 99.6%

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

      7. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. fma-undefine 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-/l/ 99.7%

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

      6. add-sqr-sqrt 47.8%

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

      7. *-un-lft-identity 47.8%

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

      8. times-frac 47.7%

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

      9. +-commutative 47.7%

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

      10. fma-undefine 47.7%

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

      11. *-commutative 47.7%

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

      12. sqrt-prod 47.7%

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

      13. fma-undefine 47.7%

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

      14. +-commutative 47.7%

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

      15. hypot-1-def 47.7%

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

      16. +-commutative 47.7%

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

   6. Applied egg-rr 47.7%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

      3. clear-num 23.1%

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

      4. sqrt-div 20.7%

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

      5. metadata-eval 20.7%

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

      6. div-inv 20.7%

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

      7. clear-num 20.7%

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

      8. /-rgt-identity 20.7%

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

      9. associate-*l* 20.7%

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

   8. Applied egg-rr 20.7%

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

   9. Taylor expanded in z around 0 99.4%

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

       1. associate-/l/ 99.5%

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

   11. Simplified 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 84.9%

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

      1. associate-/l/ 84.4%

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

      2. associate-*l* 85.1%

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

      3. *-commutative 85.1%

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

      4. sqr-neg 85.1%

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

      5. +-commutative 85.1%

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

      6. sqr-neg 85.1%

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

      7. fma-define 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in z around inf 82.0%

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

      1. associate-/r* 82.5%

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

      2. associate-/r* 82.3%

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

   7. Simplified 82.3%

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

      1. *-un-lft-identity 82.3%

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

      2. unpow2 82.3%

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

      3. times-frac 89.7%

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

   9. Applied egg-rr 89.7%

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

       1. *-commutative 89.7%

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

       2. clear-num 89.7%

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

       3. frac-times 89.3%

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

       4. metadata-eval 89.3%

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

       5. associate-/l/ 89.2%

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

       6. associate-/r/ 89.5%

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

       7. /-rgt-identity 89.5%

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

   11. Applied egg-rr 89.5%

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

   12. Taylor expanded in z around 0 95.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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(Float64(1.0 / 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[(N[(1.0 / y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

   1. *-commutative 92.4%

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

   2. associate-*r* 92.0%

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

   3. fma-undefine 92.0%

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

   4. +-commutative 92.0%

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

   5. associate-/l/ 92.4%

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

   6. add-sqr-sqrt 45.3%

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

   7. *-un-lft-identity 45.3%

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

   8. times-frac 45.2%

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

   9. +-commutative 45.2%

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

   10. fma-undefine 45.2%

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

   11. *-commutative 45.2%

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

   12. sqrt-prod 45.2%

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

   13. fma-undefine 45.2%

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

   14. +-commutative 45.2%

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

   15. hypot-1-def 45.3%

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

   16. +-commutative 45.3%

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

6. Applied egg-rr 47.7%

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

   1. add-sqr-sqrt 26.0%

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

   2. pow2 26.0%

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

   3. clear-num 26.0%

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

   4. sqrt-div 21.7%

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

   5. metadata-eval 21.7%

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

   6. div-inv 21.7%

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

   7. clear-num 21.7%

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

   8. /-rgt-identity 21.7%

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

   9. associate-*l* 21.7%

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

8. Applied egg-rr 21.7%

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

9. Taylor expanded in z around 0 58.1%

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

    1. associate-/l/ 58.2%

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

11. Simplified 58.2%

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

Alternative 10: 58.0% accurate, 2.2× speedup

Math

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

FPCore

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

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 / x) / y_m)
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 / x) / 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 * ((1.0 / x) / 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(1.0 / x) / 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 * ((1.0 / x) / 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[(1.0 / x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

5. Taylor expanded in z around 0 58.1%

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

   1. associate-/r* 58.2%

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

7. Simplified 58.2%

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

Alternative 11: 58.2% 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 #s(literal 1 binary64) 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 92.4%

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

   1. associate-/l/ 92.0%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

5. Taylor expanded in z around 0 58.1%

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

Developer Target 1: 92.5% 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 2024146 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

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