Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 68.8% → 90.2%

Time: 12.8s

Alternatives: 7

Speedup: 0.9×

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.35 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m + z\_m}{y\_m} \cdot \frac{x\_m - z\_m}{2}\\ \mathbf{elif}\;y\_m \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z\_m}{\frac{y\_m}{z\_m}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.35e-75)
    (* (/ (+ x_m z_m) y_m) (/ (- x_m z_m) 2.0))
    (if (<= y_m 2e+143)
      (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))
      (* 0.5 (- y_m (/ z_m (/ y_m z_m))))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2.35e-75) {
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	} else if (y_m <= 2e+143) {
		tmp = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 2.35d-75) then
        tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0d0)
    else if (y_m <= 2d+143) then
        tmp = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z_m / (y_m / z_m)))
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 2.35e-75) {
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	} else if (y_m <= 2e+143) {
		tmp = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 2.35e-75:
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0)
	elif y_m <= 2e+143:
		tmp = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)))
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 2.35e-75)
		tmp = Float64(Float64(Float64(x_m + z_m) / y_m) * Float64(Float64(x_m - z_m) / 2.0));
	elseif (y_m <= 2e+143)
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z_m / Float64(y_m / z_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 2.35e-75)
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	elseif (y_m <= 2e+143)
		tmp = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 2.35e-75], N[(N[(N[(x$95$m + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2e+143], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z$95$m / N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.3499999999999999e-75

   1. Initial program 75.1%

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

      1. add-cube-cbrt 74.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x \cdot x + y \cdot y} \cdot \sqrt[3]{x \cdot x + y \cdot y}\right) \cdot \sqrt[3]{x \cdot x + y \cdot y}} - z \cdot z}{y \cdot 2} \]

      2. pow3 74.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x \cdot x + y \cdot y}\right)}^{3}} - z \cdot z}{y \cdot 2} \]

      3. add-sqr-sqrt 74.8%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

      4. pow2 74.8%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{{\left(\sqrt{x \cdot x + y \cdot y}\right)}^{2}}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

      5. hypot-def 74.8%

         \[\leadsto \frac{{\left(\sqrt[3]{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 74.8%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}\right)}^{3}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in y around 0 65.9%

      \[\leadsto \frac{{\color{blue}{\left({\left({x}^{2}\right)}^{0.3333333333333333}\right)}}^{3} - z \cdot z}{y \cdot 2} \]
   6. Step-by-step derivation

      1. unpow1/3 66.8%

         \[\leadsto \frac{{\color{blue}{\left(\sqrt[3]{{x}^{2}}\right)}}^{3} - z \cdot z}{y \cdot 2} \]

   7. Simplified 66.8%

      \[\leadsto \frac{{\color{blue}{\left(\sqrt[3]{{x}^{2}}\right)}}^{3} - z \cdot z}{y \cdot 2} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 66.9%

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

      2. unpow2 66.9%

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

   9. Applied egg-rr 66.9%

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

       1. difference-of-squares 71.4%

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

       2. times-frac 75.8%

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

   11. Applied egg-rr 75.8%

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

   if 2.3499999999999999e-75 < y < 2e143

   1. Initial program 95.0%

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

   if 2e143 < y

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0 7.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. div-sub 7.7%

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

      2. unpow2 7.7%

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

      3. associate-/l* 74.2%

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

      4. *-inverses 74.2%

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

      5. /-rgt-identity 74.2%

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

   5. Simplified 74.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. pow2 74.2%

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

      2. *-un-lft-identity 74.2%

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

      3. times-frac 91.5%

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

   7. Applied egg-rr 91.5%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
   8. Step-by-step derivation

      1. /-rgt-identity 91.5%

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

      2. clear-num 91.5%

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

      3. un-div-inv 91.5%

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

   9. Applied egg-rr 91.5%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{\frac{y}{z}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-75}:\\ \;\;\;\;\frac{x + z}{y} \cdot \frac{x - z}{2}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y\_m - \frac{z\_m}{\frac{y\_m}{z\_m}}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 62000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 3550000000000:\\ \;\;\;\;\frac{x\_m \cdot 0.5}{\frac{y\_m}{x\_m}}\\ \mathbf{elif}\;x\_m \leq 2.35 \cdot 10^{+60} \lor \neg \left(x\_m \leq 3.3 \cdot 10^{+153}\right) \land x\_m \leq 3.5 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (- y_m (/ z_m (/ y_m z_m))))))
   (*
    y_s
    (if (<= x_m 62000000.0)
      t_0
      (if (<= x_m 3550000000000.0)
        (/ (* x_m 0.5) (/ y_m x_m))
        (if (or (<= x_m 2.35e+60)
                (and (not (<= x_m 3.3e+153)) (<= x_m 3.5e+162)))
          t_0
          (* (/ x_m y_m) (/ x_m 2.0))))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	double tmp;
	if (x_m <= 62000000.0) {
		tmp = t_0;
	} else if (x_m <= 3550000000000.0) {
		tmp = (x_m * 0.5) / (y_m / x_m);
	} else if ((x_m <= 2.35e+60) || (!(x_m <= 3.3e+153) && (x_m <= 3.5e+162))) {
		tmp = t_0;
	} else {
		tmp = (x_m / y_m) * (x_m / 2.0);
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (y_m - (z_m / (y_m / z_m)))
    if (x_m <= 62000000.0d0) then
        tmp = t_0
    else if (x_m <= 3550000000000.0d0) then
        tmp = (x_m * 0.5d0) / (y_m / x_m)
    else if ((x_m <= 2.35d+60) .or. (.not. (x_m <= 3.3d+153)) .and. (x_m <= 3.5d+162)) then
        tmp = t_0
    else
        tmp = (x_m / y_m) * (x_m / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	double tmp;
	if (x_m <= 62000000.0) {
		tmp = t_0;
	} else if (x_m <= 3550000000000.0) {
		tmp = (x_m * 0.5) / (y_m / x_m);
	} else if ((x_m <= 2.35e+60) || (!(x_m <= 3.3e+153) && (x_m <= 3.5e+162))) {
		tmp = t_0;
	} else {
		tmp = (x_m / y_m) * (x_m / 2.0);
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)))
	tmp = 0
	if x_m <= 62000000.0:
		tmp = t_0
	elif x_m <= 3550000000000.0:
		tmp = (x_m * 0.5) / (y_m / x_m)
	elif (x_m <= 2.35e+60) or (not (x_m <= 3.3e+153) and (x_m <= 3.5e+162)):
		tmp = t_0
	else:
		tmp = (x_m / y_m) * (x_m / 2.0)
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(0.5 * Float64(y_m - Float64(z_m / Float64(y_m / z_m))))
	tmp = 0.0
	if (x_m <= 62000000.0)
		tmp = t_0;
	elseif (x_m <= 3550000000000.0)
		tmp = Float64(Float64(x_m * 0.5) / Float64(y_m / x_m));
	elseif ((x_m <= 2.35e+60) || (!(x_m <= 3.3e+153) && (x_m <= 3.5e+162)))
		tmp = t_0;
	else
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	tmp = 0.0;
	if (x_m <= 62000000.0)
		tmp = t_0;
	elseif (x_m <= 3550000000000.0)
		tmp = (x_m * 0.5) / (y_m / x_m);
	elseif ((x_m <= 2.35e+60) || (~((x_m <= 3.3e+153)) && (x_m <= 3.5e+162)))
		tmp = t_0;
	else
		tmp = (x_m / y_m) * (x_m / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(0.5 * N[(y$95$m - N[(z$95$m / N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x$95$m, 62000000.0], t$95$0, If[LessEqual[x$95$m, 3550000000000.0], N[(N[(x$95$m * 0.5), $MachinePrecision] / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$95$m, 2.35e+60], And[N[Not[LessEqual[x$95$m, 3.3e+153]], $MachinePrecision], LessEqual[x$95$m, 3.5e+162]]], t$95$0, N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / 2.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 6.2e7 or 3.55e12 < x < 2.3499999999999999e60 or 3.29999999999999994e153 < x < 3.50000000000000018e162

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0 46.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. div-sub 46.6%

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

      2. unpow2 46.6%

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

      3. associate-/l* 65.4%

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

      4. *-inverses 65.4%

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

      5. /-rgt-identity 65.4%

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

   5. Simplified 65.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. pow2 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. times-frac 72.1%

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

   7. Applied egg-rr 72.1%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
   8. Step-by-step derivation

      1. /-rgt-identity 72.1%

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

      2. clear-num 72.1%

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

      3. un-div-inv 72.1%

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

   9. Applied egg-rr 72.1%

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

   if 6.2e7 < x < 3.55e12

   1. Initial program 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. *-commutative 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. times-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. add-sqr-sqrt 99.2%

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

      8. pow2 99.2%

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

      9. hypot-def 99.2%

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

      10. pow2 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-*r/ 99.2%

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

      3. *-commutative 99.2%

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

      4. clear-num 99.2%

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

      5. rem-cube-cbrt 99.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}{y \cdot 2} \]

      6. sqr-pow 98.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot 2} \]

      7. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2}} \]

      8. sqrt-pow1 100.0%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      9. rem-cube-cbrt 100.0%

         \[\leadsto \frac{\sqrt{\color{blue}{{x}^{2}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      10. unpow2 100.0%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      11. sqrt-prod 100.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      12. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\color{blue}{x}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      13. sqrt-pow1 99.2%

          \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{2} \]

      14. rem-cube-cbrt 99.2%

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

      15. unpow2 99.2%

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

      16. sqrt-prod 99.2%

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

      17. add-sqr-sqrt 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 100.0%

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

      4. div-inv 100.0%

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

      5. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2.3499999999999999e60 < x < 3.29999999999999994e153 or 3.50000000000000018e162 < x

   1. Initial program 60.0%

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

      1. clear-num 59.9%

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

      2. inv-pow 59.9%

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

      3. *-commutative 59.9%

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

      4. *-un-lft-identity 59.9%

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

      5. times-frac 59.9%

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

      6. metadata-eval 59.9%

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

      7. add-sqr-sqrt 59.9%

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

      8. pow2 59.9%

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

      9. hypot-def 59.9%

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

      10. pow2 59.9%

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in x around inf 56.0%

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

      1. unpow-1 56.0%

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

      2. associate-*r/ 56.0%

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

      3. *-commutative 56.0%

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

      4. clear-num 56.1%

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

      5. rem-cube-cbrt 56.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}{y \cdot 2} \]

      6. sqr-pow 56.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot 2} \]

      7. times-frac 56.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2}} \]

      8. sqrt-pow1 56.0%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      9. rem-cube-cbrt 56.0%

         \[\leadsto \frac{\sqrt{\color{blue}{{x}^{2}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      10. unpow2 56.0%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      11. sqrt-prod 56.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      12. add-sqr-sqrt 56.0%

          \[\leadsto \frac{\color{blue}{x}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      13. sqrt-pow1 56.0%

          \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{2} \]

      14. rem-cube-cbrt 56.1%

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

      15. unpow2 56.1%

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

      16. sqrt-prod 60.1%

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

      17. add-sqr-sqrt 60.1%

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

   7. Applied egg-rr 60.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 62000000:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \leq 3550000000000:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+60} \lor \neg \left(x \leq 3.3 \cdot 10^{+153}\right) \land x \leq 3.5 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y\_m - \frac{z\_m}{\frac{y\_m}{z\_m}}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 62000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 3550000000000:\\ \;\;\;\;\frac{x\_m \cdot 0.5}{\frac{y\_m}{x\_m}}\\ \mathbf{elif}\;x\_m \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z\_m \cdot \frac{z\_m}{y\_m}\right)\\ \mathbf{elif}\;x\_m \leq 3.3 \cdot 10^{+153} \lor \neg \left(x\_m \leq 1.35 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \frac{x\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (- y_m (/ z_m (/ y_m z_m))))))
   (*
    y_s
    (if (<= x_m 62000000.0)
      t_0
      (if (<= x_m 3550000000000.0)
        (/ (* x_m 0.5) (/ y_m x_m))
        (if (<= x_m 1.65e+60)
          (* 0.5 (- y_m (* z_m (/ z_m y_m))))
          (if (or (<= x_m 3.3e+153) (not (<= x_m 1.35e+160)))
            (* (/ x_m y_m) (/ x_m 2.0))
            t_0)))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	double tmp;
	if (x_m <= 62000000.0) {
		tmp = t_0;
	} else if (x_m <= 3550000000000.0) {
		tmp = (x_m * 0.5) / (y_m / x_m);
	} else if (x_m <= 1.65e+60) {
		tmp = 0.5 * (y_m - (z_m * (z_m / y_m)));
	} else if ((x_m <= 3.3e+153) || !(x_m <= 1.35e+160)) {
		tmp = (x_m / y_m) * (x_m / 2.0);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (y_m - (z_m / (y_m / z_m)))
    if (x_m <= 62000000.0d0) then
        tmp = t_0
    else if (x_m <= 3550000000000.0d0) then
        tmp = (x_m * 0.5d0) / (y_m / x_m)
    else if (x_m <= 1.65d+60) then
        tmp = 0.5d0 * (y_m - (z_m * (z_m / y_m)))
    else if ((x_m <= 3.3d+153) .or. (.not. (x_m <= 1.35d+160))) then
        tmp = (x_m / y_m) * (x_m / 2.0d0)
    else
        tmp = t_0
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	double tmp;
	if (x_m <= 62000000.0) {
		tmp = t_0;
	} else if (x_m <= 3550000000000.0) {
		tmp = (x_m * 0.5) / (y_m / x_m);
	} else if (x_m <= 1.65e+60) {
		tmp = 0.5 * (y_m - (z_m * (z_m / y_m)));
	} else if ((x_m <= 3.3e+153) || !(x_m <= 1.35e+160)) {
		tmp = (x_m / y_m) * (x_m / 2.0);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)))
	tmp = 0
	if x_m <= 62000000.0:
		tmp = t_0
	elif x_m <= 3550000000000.0:
		tmp = (x_m * 0.5) / (y_m / x_m)
	elif x_m <= 1.65e+60:
		tmp = 0.5 * (y_m - (z_m * (z_m / y_m)))
	elif (x_m <= 3.3e+153) or not (x_m <= 1.35e+160):
		tmp = (x_m / y_m) * (x_m / 2.0)
	else:
		tmp = t_0
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(0.5 * Float64(y_m - Float64(z_m / Float64(y_m / z_m))))
	tmp = 0.0
	if (x_m <= 62000000.0)
		tmp = t_0;
	elseif (x_m <= 3550000000000.0)
		tmp = Float64(Float64(x_m * 0.5) / Float64(y_m / x_m));
	elseif (x_m <= 1.65e+60)
		tmp = Float64(0.5 * Float64(y_m - Float64(z_m * Float64(z_m / y_m))));
	elseif ((x_m <= 3.3e+153) || !(x_m <= 1.35e+160))
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m / 2.0));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = 0.5 * (y_m - (z_m / (y_m / z_m)));
	tmp = 0.0;
	if (x_m <= 62000000.0)
		tmp = t_0;
	elseif (x_m <= 3550000000000.0)
		tmp = (x_m * 0.5) / (y_m / x_m);
	elseif (x_m <= 1.65e+60)
		tmp = 0.5 * (y_m - (z_m * (z_m / y_m)));
	elseif ((x_m <= 3.3e+153) || ~((x_m <= 1.35e+160)))
		tmp = (x_m / y_m) * (x_m / 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(0.5 * N[(y$95$m - N[(z$95$m / N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x$95$m, 62000000.0], t$95$0, If[LessEqual[x$95$m, 3550000000000.0], N[(N[(x$95$m * 0.5), $MachinePrecision] / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.65e+60], N[(0.5 * N[(y$95$m - N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$95$m, 3.3e+153], N[Not[LessEqual[x$95$m, 1.35e+160]], $MachinePrecision]], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if x < 6.2e7 or 3.29999999999999994e153 < x < 1.35e160

   1. Initial program 69.1%

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

   3. Taylor expanded in x around 0 46.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. div-sub 46.9%

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

      2. unpow2 46.9%

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

      3. associate-/l* 66.1%

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

      4. *-inverses 66.1%

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

      5. /-rgt-identity 66.1%

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

   5. Simplified 66.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. pow2 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. times-frac 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
   8. Step-by-step derivation

      1. /-rgt-identity 73.2%

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

      2. clear-num 73.2%

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

      3. un-div-inv 73.2%

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

   9. Applied egg-rr 73.2%

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

   if 6.2e7 < x < 3.55e12

   1. Initial program 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. *-commutative 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. times-frac 99.2%

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

      6. metadata-eval 99.2%

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

      7. add-sqr-sqrt 99.2%

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

      8. pow2 99.2%

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

      9. hypot-def 99.2%

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

      10. pow2 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-*r/ 99.2%

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

      3. *-commutative 99.2%

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

      4. clear-num 99.2%

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

      5. rem-cube-cbrt 99.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}{y \cdot 2} \]

      6. sqr-pow 98.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot 2} \]

      7. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2}} \]

      8. sqrt-pow1 100.0%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      9. rem-cube-cbrt 100.0%

         \[\leadsto \frac{\sqrt{\color{blue}{{x}^{2}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      10. unpow2 100.0%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      11. sqrt-prod 100.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      12. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\color{blue}{x}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      13. sqrt-pow1 99.2%

          \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{2} \]

      14. rem-cube-cbrt 99.2%

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

      15. unpow2 99.2%

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

      16. sqrt-prod 99.2%

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

      17. add-sqr-sqrt 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 100.0%

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

      4. div-inv 100.0%

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

      5. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 3.55e12 < x < 1.6499999999999999e60

   1. Initial program 90.5%

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

   3. Taylor expanded in x around 0 41.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. div-sub 41.4%

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

      2. unpow2 41.4%

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

      3. associate-/l* 50.9%

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

      4. *-inverses 50.9%

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

      5. /-rgt-identity 50.9%

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

   5. Simplified 50.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. pow2 50.9%

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

      2. *-un-lft-identity 50.9%

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

      3. times-frac 50.9%

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

   7. Applied egg-rr 50.9%

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

   if 1.6499999999999999e60 < x < 3.29999999999999994e153 or 1.35e160 < x

   1. Initial program 60.0%

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

      1. clear-num 59.9%

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

      2. inv-pow 59.9%

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

      3. *-commutative 59.9%

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

      4. *-un-lft-identity 59.9%

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

      5. times-frac 59.9%

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

      6. metadata-eval 59.9%

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

      7. add-sqr-sqrt 59.9%

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

      8. pow2 59.9%

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

      9. hypot-def 59.9%

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

      10. pow2 59.9%

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in x around inf 56.0%

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

      1. unpow-1 56.0%

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

      2. associate-*r/ 56.0%

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

      3. *-commutative 56.0%

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

      4. clear-num 56.1%

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

      5. rem-cube-cbrt 56.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}{y \cdot 2} \]

      6. sqr-pow 56.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot 2} \]

      7. times-frac 56.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2}} \]

      8. sqrt-pow1 56.0%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      9. rem-cube-cbrt 56.0%

         \[\leadsto \frac{\sqrt{\color{blue}{{x}^{2}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      10. unpow2 56.0%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      11. sqrt-prod 56.0%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      12. add-sqr-sqrt 56.0%

          \[\leadsto \frac{\color{blue}{x}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      13. sqrt-pow1 56.0%

          \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{2} \]

      14. rem-cube-cbrt 56.1%

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

      15. unpow2 56.1%

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

      16. sqrt-prod 60.1%

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

      17. add-sqr-sqrt 60.1%

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

   7. Applied egg-rr 60.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 62000000:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;x \leq 3550000000000:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+153} \lor \neg \left(x \leq 1.35 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m} \cdot \frac{x\_m}{2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z\_m \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z\_m \leq 3.55 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z\_m \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{elif}\;z\_m \leq 9.6 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \frac{-0.5}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (/ x_m y_m) (/ x_m 2.0))))
   (*
    y_s
    (if (<= z_m 1.25e-216)
      t_0
      (if (<= z_m 1.02e-156)
        (* y_m 0.5)
        (if (<= z_m 3.55e-107)
          t_0
          (if (<= z_m 9.2e-63)
            (* y_m 0.5)
            (if (<= z_m 9.6e+140) t_0 (* z_m (* z_m (/ -0.5 y_m)))))))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (x_m / y_m) * (x_m / 2.0);
	double tmp;
	if (z_m <= 1.25e-216) {
		tmp = t_0;
	} else if (z_m <= 1.02e-156) {
		tmp = y_m * 0.5;
	} else if (z_m <= 3.55e-107) {
		tmp = t_0;
	} else if (z_m <= 9.2e-63) {
		tmp = y_m * 0.5;
	} else if (z_m <= 9.6e+140) {
		tmp = t_0;
	} else {
		tmp = z_m * (z_m * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m / y_m) * (x_m / 2.0d0)
    if (z_m <= 1.25d-216) then
        tmp = t_0
    else if (z_m <= 1.02d-156) then
        tmp = y_m * 0.5d0
    else if (z_m <= 3.55d-107) then
        tmp = t_0
    else if (z_m <= 9.2d-63) then
        tmp = y_m * 0.5d0
    else if (z_m <= 9.6d+140) then
        tmp = t_0
    else
        tmp = z_m * (z_m * ((-0.5d0) / y_m))
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (x_m / y_m) * (x_m / 2.0);
	double tmp;
	if (z_m <= 1.25e-216) {
		tmp = t_0;
	} else if (z_m <= 1.02e-156) {
		tmp = y_m * 0.5;
	} else if (z_m <= 3.55e-107) {
		tmp = t_0;
	} else if (z_m <= 9.2e-63) {
		tmp = y_m * 0.5;
	} else if (z_m <= 9.6e+140) {
		tmp = t_0;
	} else {
		tmp = z_m * (z_m * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = (x_m / y_m) * (x_m / 2.0)
	tmp = 0
	if z_m <= 1.25e-216:
		tmp = t_0
	elif z_m <= 1.02e-156:
		tmp = y_m * 0.5
	elif z_m <= 3.55e-107:
		tmp = t_0
	elif z_m <= 9.2e-63:
		tmp = y_m * 0.5
	elif z_m <= 9.6e+140:
		tmp = t_0
	else:
		tmp = z_m * (z_m * (-0.5 / y_m))
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(x_m / y_m) * Float64(x_m / 2.0))
	tmp = 0.0
	if (z_m <= 1.25e-216)
		tmp = t_0;
	elseif (z_m <= 1.02e-156)
		tmp = Float64(y_m * 0.5);
	elseif (z_m <= 3.55e-107)
		tmp = t_0;
	elseif (z_m <= 9.2e-63)
		tmp = Float64(y_m * 0.5);
	elseif (z_m <= 9.6e+140)
		tmp = t_0;
	else
		tmp = Float64(z_m * Float64(z_m * Float64(-0.5 / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = (x_m / y_m) * (x_m / 2.0);
	tmp = 0.0;
	if (z_m <= 1.25e-216)
		tmp = t_0;
	elseif (z_m <= 1.02e-156)
		tmp = y_m * 0.5;
	elseif (z_m <= 3.55e-107)
		tmp = t_0;
	elseif (z_m <= 9.2e-63)
		tmp = y_m * 0.5;
	elseif (z_m <= 9.6e+140)
		tmp = t_0;
	else
		tmp = z_m * (z_m * (-0.5 / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z$95$m, 1.25e-216], t$95$0, If[LessEqual[z$95$m, 1.02e-156], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z$95$m, 3.55e-107], t$95$0, If[LessEqual[z$95$m, 9.2e-63], N[(y$95$m * 0.5), $MachinePrecision], If[LessEqual[z$95$m, 9.6e+140], t$95$0, N[(z$95$m * N[(z$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.25000000000000005e-216 or 1.02e-156 < z < 3.5499999999999998e-107 or 9.2e-63 < z < 9.5999999999999999e140

   1. Initial program 72.1%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

      3. *-commutative 72.0%

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

      4. *-un-lft-identity 72.0%

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

      5. times-frac 72.0%

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

      6. metadata-eval 72.0%

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

      7. add-sqr-sqrt 72.0%

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

      8. pow2 72.0%

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

      9. hypot-def 72.0%

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

      10. pow2 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in x around inf 38.0%

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

      1. unpow-1 38.0%

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

      2. associate-*r/ 38.0%

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

      3. *-commutative 38.0%

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

      4. clear-num 38.0%

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

      5. rem-cube-cbrt 37.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}{y \cdot 2} \]

      6. sqr-pow 37.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}}{y \cdot 2} \]

      7. times-frac 37.9%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2}} \]

      8. sqrt-pow1 37.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      9. rem-cube-cbrt 37.9%

         \[\leadsto \frac{\sqrt{\color{blue}{{x}^{2}}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      10. unpow2 37.9%

          \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      11. sqrt-prod 16.7%

          \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      12. add-sqr-sqrt 20.0%

          \[\leadsto \frac{\color{blue}{x}}{y} \cdot \frac{{\left(\sqrt[3]{{x}^{2}}\right)}^{\left(\frac{3}{2}\right)}}{2} \]

      13. sqrt-pow1 20.0%

          \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{{x}^{2}}\right)}^{3}}}}{2} \]

      14. rem-cube-cbrt 20.0%

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

      15. unpow2 20.0%

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

      16. sqrt-prod 18.0%

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

      17. add-sqr-sqrt 40.7%

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

   7. Applied egg-rr 40.7%

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

   if 1.25000000000000005e-216 < z < 1.02e-156 or 3.5499999999999998e-107 < z < 9.2e-63

   1. Initial program 51.7%

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

   3. Taylor expanded in y around inf 65.7%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 9.5999999999999999e140 < z

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf 75.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. associate-/l* 75.3%

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

   5. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 75.3%

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

      2. pow2 75.3%

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

      3. associate-*r* 77.8%

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

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{-216}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 3.55 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-63}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{x\_m + z\_m}{y\_m} \cdot \frac{x\_m - z\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z\_m}{\frac{y\_m}{z\_m}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.52e+29)
    (* (/ (+ x_m z_m) y_m) (/ (- x_m z_m) 2.0))
    (* 0.5 (- y_m (/ z_m (/ y_m z_m)))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.52e+29) {
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	} else {
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.52d+29) then
        tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z_m / (y_m / z_m)))
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.52e+29) {
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	} else {
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 1.52e+29:
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0)
	else:
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)))
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.52e+29)
		tmp = Float64(Float64(Float64(x_m + z_m) / y_m) * Float64(Float64(x_m - z_m) / 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z_m / Float64(y_m / z_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.52e+29)
		tmp = ((x_m + z_m) / y_m) * ((x_m - z_m) / 2.0);
	else
		tmp = 0.5 * (y_m - (z_m / (y_m / z_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 1.52e+29], N[(N[(N[(x$95$m + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z$95$m / N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.52e29

   1. Initial program 77.3%

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

      1. add-cube-cbrt 77.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x \cdot x + y \cdot y} \cdot \sqrt[3]{x \cdot x + y \cdot y}\right) \cdot \sqrt[3]{x \cdot x + y \cdot y}} - z \cdot z}{y \cdot 2} \]

      2. pow3 77.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x \cdot x + y \cdot y}\right)}^{3}} - z \cdot z}{y \cdot 2} \]

      3. add-sqr-sqrt 77.0%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

      4. pow2 77.0%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{{\left(\sqrt{x \cdot x + y \cdot y}\right)}^{2}}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

      5. hypot-def 77.0%

         \[\leadsto \frac{{\left(\sqrt[3]{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2}}\right)}^{3} - z \cdot z}{y \cdot 2} \]

   4. Applied egg-rr 77.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}\right)}^{3}} - z \cdot z}{y \cdot 2} \]

   5. Taylor expanded in y around 0 67.4%

      \[\leadsto \frac{{\color{blue}{\left({\left({x}^{2}\right)}^{0.3333333333333333}\right)}}^{3} - z \cdot z}{y \cdot 2} \]
   6. Step-by-step derivation

      1. unpow1/3 68.3%

         \[\leadsto \frac{{\color{blue}{\left(\sqrt[3]{{x}^{2}}\right)}}^{3} - z \cdot z}{y \cdot 2} \]

   7. Simplified 68.3%

      \[\leadsto \frac{{\color{blue}{\left(\sqrt[3]{{x}^{2}}\right)}}^{3} - z \cdot z}{y \cdot 2} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 68.4%

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

      2. unpow2 68.4%

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

   9. Applied egg-rr 68.4%

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

       1. difference-of-squares 72.5%

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

       2. times-frac 76.5%

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

   11. Applied egg-rr 76.5%

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

   if 1.52e29 < y

   1. Initial program 36.8%

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

   3. Taylor expanded in x around 0 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. div-sub 29.1%

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

      2. unpow2 29.1%

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

      3. associate-/l* 72.2%

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

      4. *-inverses 72.2%

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

      5. /-rgt-identity 72.2%

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

   5. Simplified 72.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. pow2 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. times-frac 85.1%

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

   7. Applied egg-rr 85.1%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
   8. Step-by-step derivation

      1. /-rgt-identity 85.1%

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

      2. clear-num 85.1%

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

      3. un-div-inv 85.1%

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

   9. Applied egg-rr 85.1%

      \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{\frac{y}{z}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{x + z}{y} \cdot \frac{x - z}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+115}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \frac{-0.5}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (* y_s (if (<= z_m 3.5e+115) (* y_m 0.5) (* z_m (* z_m (/ -0.5 y_m))))))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.5e+115) {
		tmp = y_m * 0.5;
	} else {
		tmp = z_m * (z_m * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 3.5d+115) then
        tmp = y_m * 0.5d0
    else
        tmp = z_m * (z_m * ((-0.5d0) / y_m))
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 3.5e+115) {
		tmp = y_m * 0.5;
	} else {
		tmp = z_m * (z_m * (-0.5 / y_m));
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	tmp = 0
	if z_m <= 3.5e+115:
		tmp = y_m * 0.5
	else:
		tmp = z_m * (z_m * (-0.5 / y_m))
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 3.5e+115)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z_m * Float64(z_m * Float64(-0.5 / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 3.5e+115)
		tmp = y_m * 0.5;
	else
		tmp = z_m * (z_m * (-0.5 / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 3.5e+115], N[(y$95$m * 0.5), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000005e115

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf 36.7%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 3.50000000000000005e115 < z

   1. Initial program 59.9%

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

   3. Taylor expanded in z around inf 73.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.3%

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

      2. associate-/l* 73.3%

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

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 73.3%

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

      2. pow2 73.3%

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

      3. associate-*r* 75.8%

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

   7. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\left(\frac{-0.5}{y} \cdot z\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+115}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_m y_m z_m) :precision binary64 (* y_s (* y_m 0.5)))

C

x_m = fabs(x);
z_m = fabs(z);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	return y_s * (y_m * 0.5);
}

Fortran

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = y_s * (y_m * 0.5d0)
end function

Java

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

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	return y_s * (y_m * 0.5)

Julia

x_m = abs(x)
z_m = abs(z)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(y_m * 0.5))
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_m, y_m, z_m)
	tmp = y_s * (y_m * 0.5);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in y around inf 34.6%

   \[\leadsto \color{blue}{0.5 \cdot y} \]

4. Final simplification 34.6%

   \[\leadsto y \cdot 0.5 \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024029 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))