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

Percentage Accurate: 90.2% → 99.4%

Time: 12.1s

Alternatives: 11

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% 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: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+259}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{x\_m \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 1e+259)
    (/ (/ 1.0 x_m) (fma (* y z) z y))
    (/ (/ (pow y -0.5) (* x_m (* (hypot 1.0 z) (sqrt y)))) (hypot 1.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 1e+259) {
		tmp = (1.0 / x_m) / fma((y * z), z, y);
	} else {
		tmp = (pow(y, -0.5) / (x_m * (hypot(1.0, z) * sqrt(y)))) / hypot(1.0, z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 1e+259)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64((y ^ -0.5) / Float64(x_m * Float64(hypot(1.0, z) * sqrt(y)))) / hypot(1.0, z));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+259], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[y, -0.5], $MachinePrecision] / N[(x$95$m * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.999999999999999e258

   1. Initial program 94.7%

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

      1. +-commutative 94.7%

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

      2. distribute-lft-in 94.7%

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

      3. associate-*r* 96.1%

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

      4. *-rgt-identity 96.1%

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

      5. fma-define 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 9.999999999999999e258 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 66.2%

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

      1. associate-/l/ 66.3%

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

      2. remove-double-neg 66.3%

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

      3. distribute-rgt-neg-out 66.3%

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

      4. distribute-rgt-neg-out 66.3%

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

      5. remove-double-neg 66.3%

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

      6. associate-*l* 75.1%

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

      7. *-commutative 75.1%

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

      8. sqr-neg 75.1%

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

      9. +-commutative 75.1%

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

      10. sqr-neg 75.1%

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

      11. fma-define 75.1%

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

   3. Simplified 75.1%

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

      1. associate-*r* 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-/r* 74.4%

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

      4. *-commutative 74.4%

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

      5. associate-/l/ 74.4%

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

      6. fma-undefine 74.4%

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

      7. +-commutative 74.4%

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

      8. associate-/r* 66.2%

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

      9. add-sqr-sqrt 34.1%

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

      10. associate-/l* 34.1%

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

      11. inv-pow 34.1%

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

      12. sqrt-pow1 34.2%

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

      13. metadata-eval 34.2%

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

      14. inv-pow 34.2%

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

      15. sqrt-pow1 34.2%

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

      16. metadata-eval 34.2%

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

      17. +-commutative 34.2%

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

      18. fma-undefine 34.2%

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

      19. *-commutative 34.2%

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

   6. Applied egg-rr 34.2%

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

      1. div-inv 34.2%

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

      2. associate-*r* 34.2%

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

      3. pow-prod-up 66.2%

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

      4. metadata-eval 66.2%

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

      5. inv-pow 66.2%

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

      6. *-commutative 66.2%

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

      7. associate-/r* 66.2%

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

      8. add-sqr-sqrt 66.2%

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

      9. associate-/r* 66.2%

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

      10. un-div-inv 66.2%

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

      11. fma-undefine 66.2%

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

      12. +-commutative 66.2%

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

      13. metadata-eval 66.2%

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

      14. rem-square-sqrt 66.2%

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

      15. hypot-undefine 66.2%

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

      16. hypot-undefine 66.2%

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

      17. frac-times 75.3%

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

      18. associate-/l/ 75.3%

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

      19. un-div-inv 75.3%

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

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{{y}^{-0.5}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\right)}}{\mathsf{hypot}\left(1, z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+259}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{x \cdot \left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (((y ^ -0.5) / (x_m * hypot(1.0, z))) / (hypot(1.0, z) * sqrt(y)));
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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(N[Power[y, -0.5], $MachinePrecision] / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. associate-/l/ 90.0%

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

   2. remove-double-neg 90.0%

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

   3. distribute-rgt-neg-out 90.0%

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

   4. distribute-rgt-neg-out 90.0%

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

   5. remove-double-neg 90.0%

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

   6. associate-*l* 91.5%

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

   7. *-commutative 91.5%

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

   8. sqr-neg 91.5%

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

   9. +-commutative 91.5%

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

   10. sqr-neg 91.5%

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

   11. fma-define 91.5%

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

3. Simplified 91.5%

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

   1. associate-*r* 90.4%

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

   2. *-commutative 90.4%

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

   3. associate-/r* 90.2%

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

   4. *-commutative 90.2%

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

   5. associate-/l/ 90.3%

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

   6. fma-undefine 90.3%

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

   7. +-commutative 90.3%

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

   8. associate-/r* 90.1%

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

   9. add-sqr-sqrt 39.8%

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

   10. associate-/l* 39.9%

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

   11. inv-pow 39.9%

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

   12. sqrt-pow1 39.9%

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

   13. metadata-eval 39.9%

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

   14. inv-pow 39.9%

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

   15. sqrt-pow1 39.9%

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

   16. metadata-eval 39.9%

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

   17. +-commutative 39.9%

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

   18. fma-undefine 39.9%

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

   19. *-commutative 39.9%

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

6. Applied egg-rr 39.9%

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

   1. div-inv 39.8%

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

   2. associate-*r* 39.8%

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

   3. pow-prod-up 89.9%

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

   4. metadata-eval 89.9%

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

   5. inv-pow 89.9%

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

   6. *-commutative 89.9%

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

   7. associate-/r* 90.6%

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

   8. add-sqr-sqrt 48.8%

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

   9. associate-/r* 48.8%

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

   10. un-div-inv 48.8%

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

   11. fma-undefine 48.8%

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

   12. +-commutative 48.8%

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

   13. metadata-eval 48.8%

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

   14. rem-square-sqrt 48.8%

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

   15. hypot-undefine 48.8%

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

   16. hypot-undefine 48.8%

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

   17. frac-times 50.3%

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

   18. associate-/l/ 50.3%

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

   19. un-div-inv 50.4%

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

8. Applied egg-rr 54.4%

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

9. Final simplification 54.4%

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

Alternative 3: 94.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{x\_m \cdot \mathsf{hypot}\left(1, z\right)}}{z \cdot \sqrt{y}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 5e+301)
    (/ (/ 1.0 x_m) (fma (* y z) z y))
    (/ (/ (pow y -0.5) (* x_m (hypot 1.0 z))) (* z (sqrt y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+301) {
		tmp = (1.0 / x_m) / fma((y * z), z, y);
	} else {
		tmp = (pow(y, -0.5) / (x_m * hypot(1.0, z))) / (z * sqrt(y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+301)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64((y ^ -0.5) / Float64(x_m * hypot(1.0, z))) / Float64(z * sqrt(y)));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[y, -0.5], $MachinePrecision] / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.0000000000000004e301

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. associate-*r* 96.1%

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

      4. *-rgt-identity 96.1%

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

      5. fma-define 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 5.0000000000000004e301 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 64.6%

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

      1. associate-/l/ 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-rgt-neg-out 64.6%

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

      4. distribute-rgt-neg-out 64.6%

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

      5. remove-double-neg 64.6%

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

      6. associate-*l* 73.9%

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

      7. *-commutative 73.9%

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

      8. sqr-neg 73.9%

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

      9. +-commutative 73.9%

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

      10. sqr-neg 73.9%

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

      11. fma-define 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r* 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-/r* 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-/l/ 73.1%

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

      6. fma-undefine 73.1%

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

      7. +-commutative 73.1%

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

      8. associate-/r* 64.6%

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

      9. add-sqr-sqrt 30.9%

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

      10. associate-/l* 30.9%

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

      11. inv-pow 30.9%

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

      12. sqrt-pow1 30.9%

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

      13. metadata-eval 30.9%

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

      14. inv-pow 30.9%

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

      15. sqrt-pow1 30.9%

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

      16. metadata-eval 30.9%

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

      17. +-commutative 30.9%

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

      18. fma-undefine 30.9%

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

      19. *-commutative 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. div-inv 30.9%

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

      2. associate-*r* 30.9%

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

      3. pow-prod-up 64.6%

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

      4. metadata-eval 64.6%

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

      5. inv-pow 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-/r* 64.6%

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

      8. add-sqr-sqrt 64.6%

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

      9. associate-/r* 64.6%

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

      10. un-div-inv 64.6%

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

      11. fma-undefine 64.6%

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

      12. +-commutative 64.6%

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

      13. metadata-eval 64.6%

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

      14. rem-square-sqrt 64.6%

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

      15. hypot-undefine 64.6%

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

      16. hypot-undefine 64.6%

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

      17. frac-times 74.1%

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

      18. associate-/l/ 74.1%

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

      19. un-div-inv 74.1%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in z around inf 82.9%

      \[\leadsto \frac{\frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\color{blue}{\sqrt{y} \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{y}^{-0.5}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{z \cdot \sqrt{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \frac{{x\_m}^{-0.5}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{t\_0}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (pow x_m -0.5) z)))
   (*
    x_s
    (if (<= (* y (+ 1.0 (* z z))) 5e+301)
      (/ (/ 1.0 x_m) (fma (* y z) z y))
      (* t_0 (/ t_0 y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = pow(x_m, -0.5) / z;
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+301) {
		tmp = (1.0 / x_m) / fma((y * z), z, y);
	} else {
		tmp = t_0 * (t_0 / y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64((x_m ^ -0.5) / z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+301)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(t_0 * Float64(t_0 / y));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Power[x$95$m, -0.5], $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.0000000000000004e301

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. associate-*r* 96.1%

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

      4. *-rgt-identity 96.1%

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

      5. fma-define 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 5.0000000000000004e301 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 64.6%

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

      1. associate-/l/ 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-rgt-neg-out 64.6%

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

      4. distribute-rgt-neg-out 64.6%

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

      5. remove-double-neg 64.6%

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

      6. associate-*l* 73.9%

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

      7. *-commutative 73.9%

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

      8. sqr-neg 73.9%

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

      9. +-commutative 73.9%

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

      10. sqr-neg 73.9%

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

      11. fma-define 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r* 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-/r* 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-/l/ 73.1%

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

      6. fma-undefine 73.1%

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

      7. +-commutative 73.1%

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

      8. associate-/r* 64.6%

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

      9. add-sqr-sqrt 30.9%

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

      10. associate-/l* 30.9%

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

      11. inv-pow 30.9%

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

      12. sqrt-pow1 30.9%

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

      13. metadata-eval 30.9%

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

      14. inv-pow 30.9%

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

      15. sqrt-pow1 30.9%

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

      16. metadata-eval 30.9%

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

      17. +-commutative 30.9%

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

      18. fma-undefine 30.9%

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

      19. *-commutative 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. div-inv 30.9%

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

      2. associate-*r* 30.9%

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

      3. pow-prod-up 64.6%

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

      4. metadata-eval 64.6%

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

      5. inv-pow 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-/r* 64.6%

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

      8. add-sqr-sqrt 64.6%

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

      9. associate-/r* 64.6%

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

      10. un-div-inv 64.6%

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

      11. fma-undefine 64.6%

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

      12. +-commutative 64.6%

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

      13. metadata-eval 64.6%

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

      14. rem-square-sqrt 64.6%

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

      15. hypot-undefine 64.6%

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

      16. hypot-undefine 64.6%

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

      17. frac-times 74.1%

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

      18. associate-/l/ 74.1%

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

      19. un-div-inv 74.1%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in z around inf 64.6%

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

       1. associate-/r* 64.6%

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

       2. *-commutative 64.6%

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

       3. associate-/r* 73.8%

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

   11. Simplified 73.8%

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

       1. add-sqr-sqrt 68.9%

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

       2. associate-/l* 68.8%

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

       3. sqrt-div 35.6%

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

       4. inv-pow 35.6%

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

       5. sqrt-pow1 35.6%

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

       6. metadata-eval 35.6%

          \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{{z}^{2}}} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

       7. sqrt-pow1 33.2%

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

       8. metadata-eval 33.2%

          \[\leadsto \frac{{x}^{-0.5}}{{z}^{\color{blue}{1}}} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

       9. pow1 33.2%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{z}} \cdot \frac{\sqrt{\frac{\frac{1}{x}}{{z}^{2}}}}{y} \]

       10. sqrt-div 33.2%

           \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \frac{\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{{z}^{2}}}}}{y} \]

       11. inv-pow 33.2%

           \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \frac{\frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{{z}^{2}}}}{y} \]

       12. sqrt-pow1 33.2%

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

       13. metadata-eval 33.2%

           \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \frac{\frac{{x}^{\color{blue}{-0.5}}}{\sqrt{{z}^{2}}}}{y} \]

       14. sqrt-pow1 47.2%

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

       15. metadata-eval 47.2%

           \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \frac{\frac{{x}^{-0.5}}{{z}^{\color{blue}{1}}}}{y} \]

       16. pow1 47.2%

           \[\leadsto \frac{{x}^{-0.5}}{z} \cdot \frac{\frac{{x}^{-0.5}}{\color{blue}{z}}}{y} \]

   13. Applied egg-rr 47.2%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{z} \cdot \frac{\frac{{x}^{-0.5}}{z}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{z} \cdot \frac{\frac{{x}^{-0.5}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * ((1.0 / (x_m * hypot(1.0, z))) * ((1.0 / y) / hypot(1.0, z)));
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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(1.0 / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. associate-/l/ 90.0%

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

   2. remove-double-neg 90.0%

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

   3. distribute-rgt-neg-out 90.0%

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

   4. distribute-rgt-neg-out 90.0%

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

   5. remove-double-neg 90.0%

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

   6. associate-*l* 91.5%

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

   7. *-commutative 91.5%

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

   8. sqr-neg 91.5%

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

   9. +-commutative 91.5%

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

   10. sqr-neg 91.5%

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

   11. fma-define 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in y around 0 90.0%

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

   1. associate-*r* 90.4%

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

   2. +-commutative 90.4%

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

   3. unpow2 90.4%

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

   4. fma-undefine 90.4%

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

   5. associate-/r* 90.2%

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

7. Simplified 90.2%

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

   1. associate-/r* 90.3%

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

   2. div-inv 90.2%

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

   3. add-sqr-sqrt 90.2%

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

   4. times-frac 91.8%

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

   5. fma-undefine 91.8%

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

   6. +-commutative 91.8%

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

   7. hypot-1-def 91.8%

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

   8. associate-/r* 91.8%

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

   9. fma-undefine 91.8%

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

   10. +-commutative 91.8%

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

   11. hypot-1-def 98.1%

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

9. Applied egg-rr 98.1%

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

10. Final simplification 98.1%

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

Alternative 6: 97.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x\_m}}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* y (+ 1.0 (* z z))) 5e+301)
    (/ (/ 1.0 x_m) (fma (* y z) z y))
    (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+301) {
		tmp = (1.0 / x_m) / fma((y * z), z, y);
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+301)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y);
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5.0000000000000004e301

   1. Initial program 94.8%

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

      1. +-commutative 94.8%

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

      2. distribute-lft-in 94.8%

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

      3. associate-*r* 96.1%

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

      4. *-rgt-identity 96.1%

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

      5. fma-define 96.1%

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

   4. Applied egg-rr 96.1%

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

   if 5.0000000000000004e301 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 64.6%

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

      1. associate-/l/ 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-rgt-neg-out 64.6%

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

      4. distribute-rgt-neg-out 64.6%

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

      5. remove-double-neg 64.6%

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

      6. associate-*l* 73.9%

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

      7. *-commutative 73.9%

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

      8. sqr-neg 73.9%

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

      9. +-commutative 73.9%

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

      10. sqr-neg 73.9%

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

      11. fma-define 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r* 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-/r* 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-/l/ 73.1%

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

      6. fma-undefine 73.1%

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

      7. +-commutative 73.1%

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

      8. associate-/r* 64.6%

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

      9. add-sqr-sqrt 30.9%

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

      10. associate-/l* 30.9%

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

      11. inv-pow 30.9%

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

      12. sqrt-pow1 30.9%

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

      13. metadata-eval 30.9%

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

      14. inv-pow 30.9%

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

      15. sqrt-pow1 30.9%

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

      16. metadata-eval 30.9%

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

      17. +-commutative 30.9%

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

      18. fma-undefine 30.9%

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

      19. *-commutative 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. div-inv 30.9%

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

      2. associate-*r* 30.9%

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

      3. pow-prod-up 64.6%

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

      4. metadata-eval 64.6%

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

      5. inv-pow 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-/r* 64.6%

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

      8. add-sqr-sqrt 64.6%

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

      9. associate-/r* 64.6%

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

      10. un-div-inv 64.6%

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

      11. fma-undefine 64.6%

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

      12. +-commutative 64.6%

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

      13. metadata-eval 64.6%

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

      14. rem-square-sqrt 64.6%

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

      15. hypot-undefine 64.6%

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

      16. hypot-undefine 64.6%

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

      17. frac-times 74.1%

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

      18. associate-/l/ 74.1%

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

      19. un-div-inv 74.1%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in z around inf 64.6%

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

       1. associate-/r* 64.6%

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

       2. *-commutative 64.6%

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

       3. associate-/r* 73.8%

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

   11. Simplified 73.8%

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

       1. *-un-lft-identity 73.8%

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

       2. unpow2 73.8%

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

       3. times-frac 89.6%

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

   13. Applied egg-rr 89.6%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+300) {
		tmp = 1.0 / (y * (x_m * fma(z, z, 1.0)));
	} else {
		tmp = (1.0 / (x_m * (y * z))) / hypot(1.0, z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+300)
		tmp = Float64(1.0 / Float64(y * Float64(x_m * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(y * z))) / hypot(1.0, z));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+300], N[(1.0 / N[(y * N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e300

   1. Initial program 95.8%

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

      1. associate-/l/ 95.7%

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

      2. remove-double-neg 95.7%

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

      3. distribute-rgt-neg-out 95.7%

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

      4. distribute-rgt-neg-out 95.7%

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

      5. remove-double-neg 95.7%

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

      6. associate-*l* 97.2%

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

      7. *-commutative 97.2%

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

      8. sqr-neg 97.2%

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

      9. +-commutative 97.2%

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

      10. sqr-neg 97.2%

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

      11. fma-define 97.2%

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

   3. Simplified 97.2%

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

   if 1.0000000000000001e300 < (*.f64 z z)

   1. Initial program 71.3%

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

      1. associate-/l/ 71.3%

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

      2. remove-double-neg 71.3%

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

      3. distribute-rgt-neg-out 71.3%

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

      4. distribute-rgt-neg-out 71.3%

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

      5. remove-double-neg 71.3%

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

      6. associate-*l* 72.9%

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

      7. *-commutative 72.9%

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

      8. sqr-neg 72.9%

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

      9. +-commutative 72.9%

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

      10. sqr-neg 72.9%

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

      11. 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* 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-/r* 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-/l/ 72.3%

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

      6. fma-undefine 72.3%

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

      7. +-commutative 72.3%

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

      8. associate-/r* 71.3%

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

      9. add-sqr-sqrt 30.8%

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

      10. associate-/l* 30.8%

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

      11. inv-pow 30.8%

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

      12. sqrt-pow1 30.8%

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

      13. metadata-eval 30.8%

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

      14. inv-pow 30.8%

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

      15. sqrt-pow1 30.8%

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

      16. metadata-eval 30.8%

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

      17. +-commutative 30.8%

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

      18. fma-undefine 30.8%

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

      19. *-commutative 30.8%

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

   6. Applied egg-rr 30.8%

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

      1. div-inv 30.8%

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

      2. associate-*r* 30.8%

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

      3. pow-prod-up 71.3%

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

      4. metadata-eval 71.3%

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

      5. inv-pow 71.3%

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

      6. *-commutative 71.3%

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

      7. associate-/r* 72.7%

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

      8. add-sqr-sqrt 35.9%

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

      9. associate-/r* 35.9%

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

      10. un-div-inv 35.9%

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

      11. fma-undefine 35.9%

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

      12. +-commutative 35.9%

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

      13. metadata-eval 35.9%

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

      14. rem-square-sqrt 35.9%

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

      15. hypot-undefine 35.9%

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

      16. hypot-undefine 35.9%

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

      17. frac-times 42.3%

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

      18. associate-/l/ 42.3%

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

      19. un-div-inv 42.3%

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

   8. Applied egg-rr 53.1%

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

   9. Taylor expanded in z around inf 80.4%

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

4. Final simplification 93.2%

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

Alternative 8: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x\_m}}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z z) 4e+34)
    (/ (/ 1.0 (* y (fma z z 1.0))) x_m)
    (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+34) {
		tmp = (1.0 / (y * fma(z, z, 1.0))) / x_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+34)
		tmp = Float64(Float64(1.0 / Float64(y * fma(z, z, 1.0))) / x_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y);
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 4e+34], N[(N[(1.0 / N[(y * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999978e34

   1. Initial program 99.7%

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

      1. associate-/l/ 99.5%

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

      2. remove-double-neg 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. distribute-rgt-neg-out 99.5%

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

      5. remove-double-neg 99.5%

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

      6. associate-*l* 99.5%

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

      7. *-commutative 99.5%

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

      8. sqr-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. sqr-neg 99.5%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-/r* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l/ 99.7%

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

      6. fma-undefine 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-/r* 99.7%

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

      9. add-sqr-sqrt 42.9%

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

      10. associate-/l* 42.9%

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

      11. inv-pow 42.9%

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

      12. sqrt-pow1 43.0%

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

      13. metadata-eval 43.0%

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

      14. inv-pow 43.0%

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

      15. sqrt-pow1 42.9%

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

      16. metadata-eval 42.9%

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

      17. +-commutative 42.9%

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

      18. fma-undefine 42.9%

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

      19. *-commutative 42.9%

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

   6. Applied egg-rr 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-commutative 42.9%

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

      3. pow-prod-up 99.7%

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

      4. metadata-eval 99.7%

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

      5. inv-pow 99.7%

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

      6. fma-undefine 99.7%

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

      7. distribute-lft-in 99.7%

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

      8. associate-*l* 99.7%

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

      9. *-commutative 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. fma-undefine 99.7%

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

      12. associate-/l/ 99.5%

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

      13. associate-/r* 99.7%

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

      14. fma-undefine 99.7%

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

      15. associate-*l* 99.7%

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

      16. *-un-lft-identity 99.7%

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

      17. *-commutative 99.7%

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

      18. distribute-lft-in 99.7%

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

      19. fma-undefine 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 3.99999999999999978e34 < (*.f64 z z)

   1. Initial program 79.8%

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

      1. associate-/l/ 79.8%

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

      2. remove-double-neg 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. distribute-rgt-neg-out 79.8%

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

      5. remove-double-neg 79.8%

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

      6. associate-*l* 83.0%

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

      7. *-commutative 83.0%

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

      8. sqr-neg 83.0%

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

      9. +-commutative 83.0%

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

      10. sqr-neg 83.0%

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

      11. fma-define 83.0%

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

   3. Simplified 83.0%

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

      1. associate-*r* 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-/r* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-/l/ 80.3%

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

      6. fma-undefine 80.3%

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

      7. +-commutative 80.3%

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

      8. associate-/r* 79.8%

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

      9. add-sqr-sqrt 36.5%

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

      10. associate-/l* 36.6%

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

      11. inv-pow 36.6%

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

      12. sqrt-pow1 36.6%

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

      13. metadata-eval 36.6%

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

      14. inv-pow 36.6%

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

      15. sqrt-pow1 36.6%

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

      16. metadata-eval 36.6%

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

      17. +-commutative 36.6%

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

      18. fma-undefine 36.6%

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

      19. *-commutative 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. div-inv 36.5%

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

      2. associate-*r* 36.5%

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

      3. pow-prod-up 79.7%

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

      4. metadata-eval 79.7%

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

      5. inv-pow 79.7%

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

      6. *-commutative 79.7%

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

      7. associate-/r* 81.0%

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

      8. add-sqr-sqrt 38.4%

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

      9. associate-/r* 38.4%

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

      10. un-div-inv 38.4%

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

      11. fma-undefine 38.4%

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

      12. +-commutative 38.4%

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

      13. metadata-eval 38.4%

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

      14. rem-square-sqrt 38.4%

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

      15. hypot-undefine 38.4%

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

      16. hypot-undefine 38.4%

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

      17. frac-times 41.5%

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

      18. associate-/l/ 41.5%

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

      19. un-div-inv 41.5%

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

   8. Applied egg-rr 49.7%

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

   9. Taylor expanded in z around inf 79.8%

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

       1. associate-/r* 79.8%

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

       2. *-commutative 79.8%

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

       3. associate-/r* 83.2%

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

   11. Simplified 83.2%

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

       1. *-un-lft-identity 83.2%

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

       2. unpow2 83.2%

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

       3. times-frac 90.4%

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

   13. Applied egg-rr 90.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x\_m}}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* z z) 4e+34)
    (/ (/ 1.0 x_m) (* y (+ 1.0 (* z z))))
    (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+34) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 4d+34) then
        tmp = (1.0d0 / x_m) / (y * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+34) {
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (z * z) <= 4e+34:
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+34)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e+34)
		tmp = (1.0 / x_m) / (y * (1.0 + (z * z)));
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	end
	tmp_2 = 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 4e+34], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999978e34

   1. Initial program 99.7%

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

   if 3.99999999999999978e34 < (*.f64 z z)

   1. Initial program 79.8%

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

      1. associate-/l/ 79.8%

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

      2. remove-double-neg 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. distribute-rgt-neg-out 79.8%

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

      5. remove-double-neg 79.8%

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

      6. associate-*l* 83.0%

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

      7. *-commutative 83.0%

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

      8. sqr-neg 83.0%

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

      9. +-commutative 83.0%

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

      10. sqr-neg 83.0%

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

      11. fma-define 83.0%

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

   3. Simplified 83.0%

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

      1. associate-*r* 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-/r* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-/l/ 80.3%

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

      6. fma-undefine 80.3%

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

      7. +-commutative 80.3%

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

      8. associate-/r* 79.8%

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

      9. add-sqr-sqrt 36.5%

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

      10. associate-/l* 36.6%

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

      11. inv-pow 36.6%

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

      12. sqrt-pow1 36.6%

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

      13. metadata-eval 36.6%

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

      14. inv-pow 36.6%

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

      15. sqrt-pow1 36.6%

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

      16. metadata-eval 36.6%

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

      17. +-commutative 36.6%

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

      18. fma-undefine 36.6%

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

      19. *-commutative 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. div-inv 36.5%

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

      2. associate-*r* 36.5%

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

      3. pow-prod-up 79.7%

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

      4. metadata-eval 79.7%

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

      5. inv-pow 79.7%

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

      6. *-commutative 79.7%

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

      7. associate-/r* 81.0%

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

      8. add-sqr-sqrt 38.4%

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

      9. associate-/r* 38.4%

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

      10. un-div-inv 38.4%

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

      11. fma-undefine 38.4%

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

      12. +-commutative 38.4%

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

      13. metadata-eval 38.4%

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

      14. rem-square-sqrt 38.4%

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

      15. hypot-undefine 38.4%

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

      16. hypot-undefine 38.4%

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

      17. frac-times 41.5%

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

      18. associate-/l/ 41.5%

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

      19. un-div-inv 41.5%

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

   8. Applied egg-rr 49.7%

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

   9. Taylor expanded in z around inf 79.8%

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

       1. associate-/r* 79.8%

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

       2. *-commutative 79.8%

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

       3. associate-/r* 83.2%

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

   11. Simplified 83.2%

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

       1. *-un-lft-identity 83.2%

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

       2. unpow2 83.2%

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

       3. times-frac 90.4%

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

   13. Applied egg-rr 90.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.7:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x\_m}}{z}}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 0.7) (/ (/ 1.0 x_m) y) (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 0.7) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.7d0) then
        tmp = (1.0d0 / x_m) / y
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 0.7) {
		tmp = (1.0 / x_m) / y;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= 0.7:
		tmp = (1.0 / x_m) / y
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 0.7)
		tmp = Float64(Float64(1.0 / x_m) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= 0.7)
		tmp = (1.0 / x_m) / y;
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y;
	end
	tmp_2 = 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, 0.7], N[(N[(1.0 / x$95$m), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.69999999999999996

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0 67.5%

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

   if 0.69999999999999996 < z

   1. Initial program 83.6%

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

      1. associate-/l/ 83.6%

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

      2. remove-double-neg 83.6%

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

      3. distribute-rgt-neg-out 83.6%

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

      4. distribute-rgt-neg-out 83.6%

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

      5. remove-double-neg 83.6%

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

      6. associate-*l* 86.7%

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

      7. *-commutative 86.7%

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

      8. sqr-neg 86.7%

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

      9. +-commutative 86.7%

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

      10. sqr-neg 86.7%

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

      11. fma-define 86.7%

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

   3. Simplified 86.7%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-/r* 83.3%

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

      4. *-commutative 83.3%

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

      5. associate-/l/ 83.2%

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

      6. fma-undefine 83.2%

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

      7. +-commutative 83.2%

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

      8. associate-/r* 83.6%

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

      9. add-sqr-sqrt 43.2%

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

      10. associate-/l* 43.3%

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

      11. inv-pow 43.3%

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

      12. sqrt-pow1 43.3%

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

      13. metadata-eval 43.3%

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

      14. inv-pow 43.3%

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

      15. sqrt-pow1 43.3%

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

      16. metadata-eval 43.3%

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

      17. +-commutative 43.3%

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

      18. fma-undefine 43.3%

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

      19. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. div-inv 43.2%

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

      2. associate-*r* 43.2%

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

      3. pow-prod-up 83.5%

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

      4. metadata-eval 83.5%

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

      5. inv-pow 83.5%

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

      6. *-commutative 83.5%

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

      7. associate-/r* 83.5%

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

      8. add-sqr-sqrt 40.3%

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

      9. associate-/r* 40.3%

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

      10. un-div-inv 40.2%

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

      11. fma-undefine 40.2%

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

      12. +-commutative 40.2%

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

      13. metadata-eval 40.2%

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

      14. rem-square-sqrt 40.2%

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

      15. hypot-undefine 40.2%

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

      16. hypot-undefine 40.3%

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

      17. frac-times 41.9%

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

      18. associate-/l/ 41.9%

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

      19. un-div-inv 42.0%

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

   8. Applied egg-rr 53.0%

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

   9. Taylor expanded in z around inf 82.9%

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

       1. associate-/r* 82.9%

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

       2. *-commutative 82.9%

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

       3. associate-/r* 86.0%

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

   11. Simplified 86.0%

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

       1. *-un-lft-identity 86.0%

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

       2. unpow2 86.0%

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

       3. times-frac 90.8%

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

   13. Applied egg-rr 90.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.7:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.1%

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

   1. associate-/l/ 90.0%

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

   2. remove-double-neg 90.0%

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

   3. distribute-rgt-neg-out 90.0%

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

   4. distribute-rgt-neg-out 90.0%

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

   5. remove-double-neg 90.0%

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

   6. associate-*l* 91.5%

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

   7. *-commutative 91.5%

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

   8. sqr-neg 91.5%

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

   9. +-commutative 91.5%

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

   10. sqr-neg 91.5%

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

   11. fma-define 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in z around 0 55.6%

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

6. Final simplification 55.6%

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

Developer target: 93.1% 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 2024131 
(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)))))