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

Percentage Accurate: 88.7% → 99.5%

Time: 11.4s

Alternatives: 11

Speedup: 0.5×

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: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \sqrt{x\_m}\\ t_1 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y\_m}}{t\_0}}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* z (sqrt x_m))) (t_1 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 5e+307) (/ (/ 1.0 x_m) t_1) (/ (/ (/ 1.0 y_m) t_0) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = z * sqrt(x_m);
	double t_1 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (1.0 / x_m) / t_1;
	} else {
		tmp = ((1.0 / y_m) / t_0) / t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sqrt(x_m)
    t_1 = y_m * (1.0d0 + (z * z))
    if (t_1 <= 5d+307) then
        tmp = (1.0d0 / x_m) / t_1
    else
        tmp = ((1.0d0 / y_m) / t_0) / t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = z * Math.sqrt(x_m);
	double t_1 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = (1.0 / x_m) / t_1;
	} else {
		tmp = ((1.0 / y_m) / t_0) / t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = z * math.sqrt(x_m)
	t_1 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_1 <= 5e+307:
		tmp = (1.0 / x_m) / t_1
	else:
		tmp = ((1.0 / y_m) / t_0) / t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(z * sqrt(x_m))
	t_1 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_1 <= 5e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / t_0) / t_0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = z * sqrt(x_m);
	t_1 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_1 <= 5e+307)
		tmp = (1.0 / x_m) / t_1;
	else
		tmp = ((1.0 / y_m) / t_0) / t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 5e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5e307

   1. Initial program 95.2%

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

   if 5e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 72.1%

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

      1. associate-/l/ 72.1%

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

      2. associate-*l* 79.0%

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

      3. *-commutative 79.0%

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

      4. sqr-neg 79.0%

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

      5. +-commutative 79.0%

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

      6. sqr-neg 79.0%

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

      7. fma-define 79.0%

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

   3. Simplified 79.0%

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

      1. associate-*r* 78.5%

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

      2. *-commutative 78.5%

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

      3. *-commutative 78.5%

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

      4. add-sqr-sqrt 71.2%

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

      5. sqrt-div 26.9%

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

      6. metadata-eval 26.9%

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

      7. sqrt-prod 26.9%

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

      8. fma-undefine 26.9%

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

      9. +-commutative 26.9%

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

      10. hypot-1-def 26.9%

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

      11. sqrt-div 26.9%

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

      12. metadata-eval 26.9%

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

      13. sqrt-prod 26.9%

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

      14. fma-undefine 26.9%

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

      15. +-commutative 26.9%

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

      16. hypot-1-def 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. unpow-1 29.5%

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

      2. unpow-1 29.5%

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

      3. pow-sqr 29.4%

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

      4. metadata-eval 29.4%

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

   8. Simplified 29.4%

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

   9. Taylor expanded in z around inf 29.4%

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

       1. metadata-eval 29.4%

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

       2. pow-sqr 29.5%

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

       3. pow-prod-down 29.4%

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

       4. *-commutative 29.4%

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

       5. *-commutative 29.4%

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

       6. swap-sqr 26.9%

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

       7. unpow2 26.9%

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

       8. add-sqr-sqrt 78.5%

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

       9. associate-*l* 79.0%

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

       10. *-commutative 79.0%

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

       11. inv-pow 79.0%

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

       12. associate-/l/ 79.0%

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

       13. add-sqr-sqrt 27.3%

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

       14. associate-/r* 27.3%

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

   11. Applied egg-rr 36.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z \cdot \sqrt{x}}}{z \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (/ (/ (/ (pow y_m -0.5) (hypot 1.0 z)) (* (hypot 1.0 z) x_m)) (sqrt y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((pow(y_m, -0.5) / hypot(1.0, z)) / (hypot(1.0, z) * x_m)) / sqrt(y_m)));
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((Math.pow(y_m, -0.5) / Math.hypot(1.0, z)) / (Math.hypot(1.0, z) * x_m)) / Math.sqrt(y_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (((math.pow(y_m, -0.5) / math.hypot(1.0, z)) / (math.hypot(1.0, z) * x_m)) / math.sqrt(y_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64((y_m ^ -0.5) / hypot(1.0, z)) / Float64(hypot(1.0, z) * x_m)) / sqrt(y_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((((y_m ^ -0.5) / hypot(1.0, z)) / (hypot(1.0, z) * x_m)) / sqrt(y_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(N[Power[y$95$m, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

   1. associate-*r* 91.5%

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

   2. *-commutative 91.5%

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

   3. *-commutative 91.5%

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

   4. add-sqr-sqrt 60.9%

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

   5. sqrt-div 43.9%

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

   6. metadata-eval 43.9%

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

   7. sqrt-prod 43.9%

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

   8. fma-undefine 43.9%

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

   9. +-commutative 43.9%

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

   10. hypot-1-def 43.9%

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

   11. sqrt-div 43.8%

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

   12. metadata-eval 43.8%

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

   13. sqrt-prod 43.8%

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

   14. fma-undefine 43.8%

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

   15. +-commutative 43.8%

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

   16. hypot-1-def 45.1%

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

6. Applied egg-rr 45.1%

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

   1. unpow-1 45.1%

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

   2. unpow-1 45.1%

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

   3. pow-sqr 45.1%

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

   4. metadata-eval 45.1%

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

8. Simplified 45.1%

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

10. Applied egg-rr 47.5%

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

11. Final simplification 47.5%

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

Alternative 3: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / y_m) * ((1.0 / hypot(1.0, z)) / (hypot(1.0, z) * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

   1. associate-*r* 91.5%

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

   2. *-commutative 91.5%

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

   3. *-commutative 91.5%

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

   4. add-sqr-sqrt 60.9%

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

   5. sqrt-div 43.9%

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

   6. metadata-eval 43.9%

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

   7. sqrt-prod 43.9%

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

   8. fma-undefine 43.9%

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

   9. +-commutative 43.9%

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

   10. hypot-1-def 43.9%

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

   11. sqrt-div 43.8%

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

   12. metadata-eval 43.8%

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

   13. sqrt-prod 43.8%

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

   14. fma-undefine 43.8%

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

   15. +-commutative 43.8%

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

   16. hypot-1-def 45.1%

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

6. Applied egg-rr 45.1%

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

   1. unpow-1 45.1%

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

   2. unpow-1 45.1%

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

   3. pow-sqr 45.1%

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

   4. metadata-eval 45.1%

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

8. Simplified 45.1%

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

10. Applied egg-rr 47.5%

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

    1. associate-/l/ 47.9%

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

    2. div-inv 47.9%

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

    3. times-frac 46.6%

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

    4. metadata-eval 46.6%

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

    5. pow-flip 46.6%

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

    6. pow1/2 46.6%

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

    7. associate-/r* 46.6%

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

    8. add-sqr-sqrt 96.3%

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

    9. *-commutative 96.3%

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

12. Applied egg-rr 96.3%

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

13. Final simplification 96.3%

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

Alternative 4: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \frac{1}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z} \cdot \frac{\frac{1}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+211)
     (* (/ 1.0 y_m) (/ 1.0 (* x_m (fma z z 1.0))))
     (* (/ (/ 1.0 y_m) z) (/ (/ 1.0 x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+211) {
		tmp = (1.0 / y_m) * (1.0 / (x_m * fma(z, z, 1.0)));
	} else {
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+211)
		tmp = Float64(Float64(1.0 / y_m) * Float64(1.0 / Float64(x_m * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z) * Float64(Float64(1.0 / x_m) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+211], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e211

   1. Initial program 97.1%

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

      1. associate-/l/ 96.6%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. sqr-neg 98.8%

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

      5. +-commutative 98.8%

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

      6. sqr-neg 98.8%

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

      7. fma-define 98.8%

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

   3. Simplified 98.8%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 1.9999999999999999e211 < (*.f64 z z)

   1. Initial program 77.8%

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

      1. associate-/l/ 77.8%

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

      2. associate-*l* 72.9%

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

      3. *-commutative 72.9%

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

      4. sqr-neg 72.9%

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

      5. +-commutative 72.9%

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

      6. sqr-neg 72.9%

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

      7. fma-define 72.9%

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

   3. Simplified 72.9%

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

      1. associate-*r* 76.1%

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

      2. *-commutative 76.1%

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

      3. *-commutative 76.1%

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

      4. add-sqr-sqrt 72.0%

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

      5. sqrt-div 33.7%

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

      6. metadata-eval 33.7%

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

      7. sqrt-prod 33.7%

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

      8. fma-undefine 33.7%

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

      9. +-commutative 33.7%

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

      10. hypot-1-def 33.7%

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

      11. sqrt-div 33.7%

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

      12. metadata-eval 33.7%

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

      13. sqrt-prod 33.7%

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

      14. fma-undefine 33.7%

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

      15. +-commutative 33.7%

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

      16. hypot-1-def 37.9%

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

   6. Applied egg-rr 37.9%

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

      1. unpow-1 37.9%

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

      2. unpow-1 37.9%

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

      3. pow-sqr 38.0%

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

      4. metadata-eval 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in z around inf 38.0%

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

       1. metadata-eval 38.0%

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

       2. pow-sqr 37.9%

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

       3. pow-prod-down 37.8%

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

       4. *-commutative 37.8%

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

       5. *-commutative 37.8%

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

       6. swap-sqr 33.8%

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

       7. unpow2 33.8%

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

       8. add-sqr-sqrt 76.1%

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

       9. associate-*l* 72.9%

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

       10. *-commutative 72.9%

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

       11. inv-pow 72.9%

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

       12. associate-/l/ 72.9%

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

       13. associate-/r* 76.0%

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

       14. associate-/r* 76.1%

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

       15. associate-/l/ 76.1%

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

       16. unpow2 76.1%

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

       17. associate-/r* 85.6%

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

       18. associate-/l/ 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. associate-/l/ 76.1%

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

       2. metadata-eval 76.1%

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

       3. frac-times 76.0%

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

       4. times-frac 99.2%

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

   13. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+211}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z} \cdot \frac{\frac{1}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+229)
     (/ 1.0 (* y_m (* x_m (fma z z 1.0))))
     (* (/ (/ 1.0 y_m) z) (/ (/ 1.0 x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+229) {
		tmp = 1.0 / (y_m * (x_m * fma(z, z, 1.0)));
	} else {
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+229)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z) * Float64(Float64(1.0 / x_m) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+229], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e229

   1. Initial program 97.1%

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

      1. associate-/l/ 96.7%

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

      2. associate-*l* 98.2%

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

      3. *-commutative 98.2%

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

      4. sqr-neg 98.2%

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

      5. +-commutative 98.2%

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

      6. sqr-neg 98.2%

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

      7. fma-define 98.2%

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

   3. Simplified 98.2%

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

   if 2e229 < (*.f64 z z)

   1. Initial program 77.2%

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

      1. associate-/l/ 77.2%

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

      2. associate-*l* 73.5%

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

      3. *-commutative 73.5%

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

      4. sqr-neg 73.5%

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

      5. +-commutative 73.5%

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

      6. sqr-neg 73.5%

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

      7. fma-define 73.5%

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

   3. Simplified 73.5%

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

      1. associate-*r* 75.4%

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

      2. *-commutative 75.4%

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

      3. *-commutative 75.4%

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

      4. add-sqr-sqrt 71.2%

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

      5. sqrt-div 31.9%

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

      6. metadata-eval 31.9%

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

      7. sqrt-prod 31.9%

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

      8. fma-undefine 31.9%

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

      9. +-commutative 31.9%

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

      10. hypot-1-def 31.9%

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

      11. sqrt-div 31.9%

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

      12. metadata-eval 31.9%

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

      13. sqrt-prod 31.9%

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

      14. fma-undefine 31.9%

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

      15. +-commutative 31.9%

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

      16. hypot-1-def 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. unpow-1 36.2%

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

      2. unpow-1 36.2%

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

      3. pow-sqr 36.3%

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

      4. metadata-eval 36.3%

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

   8. Simplified 36.3%

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

   9. Taylor expanded in z around inf 36.3%

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

       1. metadata-eval 36.3%

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

       2. pow-sqr 36.2%

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

       3. pow-prod-down 36.1%

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

       4. *-commutative 36.1%

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

       5. *-commutative 36.1%

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

       6. swap-sqr 31.9%

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

       7. unpow2 31.9%

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

       8. add-sqr-sqrt 75.4%

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

       9. associate-*l* 73.5%

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

       10. *-commutative 73.5%

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

       11. inv-pow 73.5%

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

       12. associate-/l/ 73.5%

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

       13. associate-/r* 75.4%

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

       14. associate-/r* 75.4%

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

       15. associate-/l/ 75.4%

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

       16. unpow2 75.4%

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

       17. associate-/r* 85.2%

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

       18. associate-/l/ 85.1%

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

   11. Applied egg-rr 85.1%

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

       1. associate-/l/ 75.4%

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

       2. metadata-eval 75.4%

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

       3. frac-times 75.4%

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

       4. times-frac 99.2%

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

   13. Applied egg-rr 99.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 5e+307)
       (/ (/ 1.0 x_m) t_0)
       (/ (/ 1.0 (* y_m (* z x_m))) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+307) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+307:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+307)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5e307

   1. Initial program 95.2%

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

   if 5e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 72.1%

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

      1. associate-/l/ 72.1%

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

      2. associate-*l* 79.0%

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

      3. *-commutative 79.0%

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

      4. sqr-neg 79.0%

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

      5. +-commutative 79.0%

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

      6. sqr-neg 79.0%

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

      7. fma-define 79.0%

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

   3. Simplified 79.0%

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

      1. associate-*r* 78.5%

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

      2. *-commutative 78.5%

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

      3. *-commutative 78.5%

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

      4. add-sqr-sqrt 71.2%

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

      5. sqrt-div 26.9%

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

      6. metadata-eval 26.9%

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

      7. sqrt-prod 26.9%

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

      8. fma-undefine 26.9%

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

      9. +-commutative 26.9%

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

      10. hypot-1-def 26.9%

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

      11. sqrt-div 26.9%

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

      12. metadata-eval 26.9%

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

      13. sqrt-prod 26.9%

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

      14. fma-undefine 26.9%

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

      15. +-commutative 26.9%

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

      16. hypot-1-def 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. unpow-1 29.5%

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

      2. unpow-1 29.5%

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

      3. pow-sqr 29.4%

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

      4. metadata-eval 29.4%

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

   8. Simplified 29.4%

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

   9. Taylor expanded in z around inf 29.4%

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

       1. metadata-eval 29.4%

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

       2. pow-sqr 29.5%

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

       3. pow-prod-down 29.4%

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

       4. *-commutative 29.4%

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

       5. *-commutative 29.4%

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

       6. swap-sqr 26.9%

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

       7. unpow2 26.9%

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

       8. add-sqr-sqrt 78.5%

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

       9. associate-*l* 79.0%

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

       10. *-commutative 79.0%

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

       11. inv-pow 79.0%

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

       12. associate-/l/ 79.0%

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

       13. associate-/r* 78.5%

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

       14. associate-/r* 78.5%

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

       15. associate-/l/ 78.5%

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

       16. unpow2 78.5%

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

       17. associate-/r* 88.1%

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

       18. associate-/l/ 88.0%

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

   11. Applied egg-rr 88.0%

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

   12. Taylor expanded in y around 0 95.2%

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

       1. *-commutative 95.2%

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

       2. associate-*r* 97.6%

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

       3. associate-/r* 97.5%

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

       4. associate-/l/ 97.6%

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

   14. Simplified 97.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{z} \cdot \frac{\frac{1}{x\_m}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 7e-5)
     (/ (/ 1.0 x_m) y_m)
     (* (/ (/ 1.0 y_m) z) (/ (/ 1.0 x_m) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = ((1.0d0 / y_m) / z) * ((1.0d0 / x_m) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 7e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 7e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / z) * Float64(Float64(1.0 / x_m) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 7e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = ((1.0 / y_m) / z) * ((1.0 / x_m) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 7e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.9999999999999994e-5

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 69.9%

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

   if 6.9999999999999994e-5 < z

   1. Initial program 82.1%

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

      1. associate-/l/ 81.8%

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

      2. associate-*l* 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. +-commutative 80.5%

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

      6. sqr-neg 80.5%

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

      7. fma-define 80.5%

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

   3. Simplified 80.5%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. add-sqr-sqrt 66.2%

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

      5. sqrt-div 37.1%

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

      6. metadata-eval 37.1%

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

      7. sqrt-prod 37.2%

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

      8. fma-undefine 37.2%

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

      9. +-commutative 37.2%

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

      10. hypot-1-def 37.2%

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

      11. sqrt-div 37.1%

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

      12. metadata-eval 37.1%

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

      13. sqrt-prod 37.3%

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

      14. fma-undefine 37.3%

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

      15. +-commutative 37.3%

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

      16. hypot-1-def 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. unpow-1 39.0%

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

      2. unpow-1 39.0%

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

      3. pow-sqr 39.1%

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

      4. metadata-eval 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in z around inf 37.5%

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

       1. metadata-eval 37.5%

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

       2. pow-sqr 37.5%

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

       3. pow-prod-down 37.3%

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

       4. *-commutative 37.3%

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

       5. *-commutative 37.3%

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

       6. swap-sqr 35.5%

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

       7. unpow2 35.5%

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

       8. add-sqr-sqrt 78.1%

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

       9. associate-*l* 78.7%

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

       10. *-commutative 78.7%

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

       11. inv-pow 78.7%

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

       12. associate-/l/ 79.1%

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

       13. associate-/r* 78.4%

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

       14. associate-/r* 78.4%

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

       15. associate-/l/ 78.4%

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

       16. unpow2 78.4%

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

       17. associate-/r* 83.3%

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

       18. associate-/l/ 83.3%

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

   11. Applied egg-rr 83.3%

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

       1. associate-/l/ 78.4%

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

       2. metadata-eval 78.4%

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

       3. frac-times 78.4%

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

       4. times-frac 95.7%

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

   13. Applied egg-rr 95.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{z \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 7e-5) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 x_m) (* z (* y_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / x_m) / (z * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / x_m) / (z * (y_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / x_m) / (z * (y_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 7e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / x_m) / (z * (y_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 7e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(z * Float64(y_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 7e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / x_m) / (z * (y_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 7e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.9999999999999994e-5

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 69.9%

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

   if 6.9999999999999994e-5 < z

   1. Initial program 82.1%

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

      1. add-sqr-sqrt 37.5%

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

      2. pow2 37.5%

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

      3. +-commutative 37.5%

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

      4. fma-undefine 37.5%

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

      5. *-commutative 37.5%

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

      6. sqrt-prod 37.5%

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

      7. fma-undefine 37.5%

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

      8. +-commutative 37.5%

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

      9. hypot-1-def 39.1%

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

   4. Applied egg-rr 39.1%

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

   5. Taylor expanded in z around inf 38.5%

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

      1. unpow2 38.5%

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

      2. associate-*r* 38.5%

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

      3. *-commutative 38.5%

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

      4. associate-*r* 38.5%

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

      5. add-sqr-sqrt 89.9%

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 74.5%

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

Alternative 9: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(y\_m \cdot z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 7e-5) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 (* x_m (* y_m z))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (x_m * (y_m * z))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / (x_m * (y_m * z))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (x_m * (y_m * z))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 7e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / (x_m * (y_m * z))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 7e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(y_m * z))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 7e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / (x_m * (y_m * z))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 7e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.9999999999999994e-5

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 69.9%

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

   if 6.9999999999999994e-5 < z

   1. Initial program 82.1%

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

      1. associate-/l/ 81.8%

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

      2. associate-*l* 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. +-commutative 80.5%

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

      6. sqr-neg 80.5%

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

      7. fma-define 80.5%

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

   3. Simplified 80.5%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. add-sqr-sqrt 66.2%

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

      5. sqrt-div 37.1%

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

      6. metadata-eval 37.1%

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

      7. sqrt-prod 37.2%

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

      8. fma-undefine 37.2%

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

      9. +-commutative 37.2%

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

      10. hypot-1-def 37.2%

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

      11. sqrt-div 37.1%

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

      12. metadata-eval 37.1%

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

      13. sqrt-prod 37.3%

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

      14. fma-undefine 37.3%

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

      15. +-commutative 37.3%

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

      16. hypot-1-def 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. unpow-1 39.0%

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

      2. unpow-1 39.0%

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

      3. pow-sqr 39.1%

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

      4. metadata-eval 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in z around inf 37.5%

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

       1. metadata-eval 37.5%

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

       2. pow-sqr 37.5%

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

       3. pow-prod-down 37.3%

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

       4. *-commutative 37.3%

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

       5. *-commutative 37.3%

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

       6. swap-sqr 35.5%

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

       7. unpow2 35.5%

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

       8. add-sqr-sqrt 78.1%

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

       9. associate-*l* 78.7%

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

       10. *-commutative 78.7%

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

       11. inv-pow 78.7%

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

       12. associate-/l/ 79.1%

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

       13. associate-/r* 78.4%

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

       14. associate-/r* 78.4%

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

       15. associate-/l/ 78.4%

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

       16. unpow2 78.4%

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

       17. associate-/r* 83.3%

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

       18. associate-/l/ 83.3%

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

   11. Applied egg-rr 83.3%

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

   12. Taylor expanded in y around 0 94.8%

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

4. Final simplification 75.6%

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

Alternative 10: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 7e-5) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 7e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 7e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 7e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 7e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 7e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.9999999999999994e-5

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0 69.9%

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

   if 6.9999999999999994e-5 < z

   1. Initial program 82.1%

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

      1. associate-/l/ 81.8%

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

      2. associate-*l* 80.5%

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

      3. *-commutative 80.5%

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

      4. sqr-neg 80.5%

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

      5. +-commutative 80.5%

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

      6. sqr-neg 80.5%

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

      7. fma-define 80.5%

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

   3. Simplified 80.5%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. add-sqr-sqrt 66.2%

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

      5. sqrt-div 37.1%

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

      6. metadata-eval 37.1%

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

      7. sqrt-prod 37.2%

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

      8. fma-undefine 37.2%

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

      9. +-commutative 37.2%

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

      10. hypot-1-def 37.2%

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

      11. sqrt-div 37.1%

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

      12. metadata-eval 37.1%

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

      13. sqrt-prod 37.3%

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

      14. fma-undefine 37.3%

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

      15. +-commutative 37.3%

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

      16. hypot-1-def 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. unpow-1 39.0%

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

      2. unpow-1 39.0%

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

      3. pow-sqr 39.1%

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

      4. metadata-eval 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in z around inf 37.5%

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

       1. metadata-eval 37.5%

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

       2. pow-sqr 37.5%

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

       3. pow-prod-down 37.3%

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

       4. *-commutative 37.3%

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

       5. *-commutative 37.3%

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

       6. swap-sqr 35.5%

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

       7. unpow2 35.5%

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

       8. add-sqr-sqrt 78.1%

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

       9. associate-*l* 78.7%

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

       10. *-commutative 78.7%

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

       11. inv-pow 78.7%

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

       12. associate-/l/ 79.1%

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

       13. associate-/r* 78.4%

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

       14. associate-/r* 78.4%

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

       15. associate-/l/ 78.4%

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

       16. unpow2 78.4%

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

       17. associate-/r* 83.3%

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

       18. associate-/l/ 83.3%

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

   11. Applied egg-rr 83.3%

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

   12. Taylor expanded in y around 0 94.8%

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

       1. *-commutative 94.8%

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

       2. associate-*r* 94.9%

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

       3. associate-/r* 94.9%

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

       4. associate-/l/ 94.9%

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

   14. Simplified 94.9%

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

4. Final simplification 75.6%

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

Alternative 11: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (y_m * x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 91.3%

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

   3. *-commutative 91.3%

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

   4. sqr-neg 91.3%

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

   5. +-commutative 91.3%

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

   6. sqr-neg 91.3%

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

   7. fma-define 91.3%

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

3. Simplified 91.3%

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

5. Taylor expanded in z around 0 59.7%

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

6. Final simplification 59.7%

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

Developer target: 92.7% 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 2024053 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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