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

Percentage Accurate: 68.9% → 94.1%

Time: 9.3s

Alternatives: 10

Speedup: 0.8×

• 41.8% 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: 68.9% 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: 94.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{+185}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= x 4.5e+185)
     (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))
     (* t_0 (* t_0 (/ 0.5 y))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if (x <= 4.5e+185) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = t_0 * (t_0 * (0.5 / y));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(y, Math.hypot(x, z));
	double tmp;
	if (x <= 4.5e+185) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = t_0 * (t_0 * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = math.hypot(y, math.hypot(x, z))
	tmp = 0
	if x <= 4.5e+185:
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)))
	else:
		tmp = t_0 * (t_0 * (0.5 / y))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (x <= 4.5e+185)
		tmp = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))));
	else
		tmp = Float64(t_0 * Float64(t_0 * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = hypot(y, hypot(x, z));
	tmp = 0.0;
	if (x <= 4.5e+185)
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	else
		tmp = t_0 * (t_0 * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[x, 4.5e+185], N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000002e185

   1. Initial program 65.4%

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

   2. Taylor expanded in x around inf 81.3%

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

      1. unpow2 81.3%

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

      2. *-un-lft-identity 81.3%

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

      3. times-frac 89.1%

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

   4. Applied egg-rr 89.1%

      \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. unpow2 89.1%

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

      2. *-un-lft-identity 89.1%

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

      3. times-frac 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. /-rgt-identity 94.0%

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

      2. associate-*r/ 89.1%

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

      3. associate-/l* 94.0%

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

   8. Applied egg-rr 94.0%

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

   if 4.5000000000000002e185 < x

   1. Initial program 80.0%

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

      1. div-inv 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. associate-*l* 80.0%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+185}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 2: 94.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;x \leq 2.2 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y} \cdot \frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= x 2.2e+158)
     (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))
     (* (/ t_0 y) (/ t_0 2.0)))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if (x <= 2.2e+158) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = (t_0 / y) * (t_0 / 2.0);
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(y, Math.hypot(x, z));
	double tmp;
	if (x <= 2.2e+158) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = (t_0 / y) * (t_0 / 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = math.hypot(y, math.hypot(x, z))
	tmp = 0
	if x <= 2.2e+158:
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)))
	else:
		tmp = (t_0 / y) * (t_0 / 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (x <= 2.2e+158)
		tmp = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))));
	else
		tmp = Float64(Float64(t_0 / y) * Float64(t_0 / 2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = hypot(y, hypot(x, z));
	tmp = 0.0;
	if (x <= 2.2e+158)
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	else
		tmp = (t_0 / y) * (t_0 / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[x, 2.2e+158], N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / y), $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000001e158

   1. Initial program 66.0%

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

   2. Taylor expanded in x around inf 82.2%

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

      1. unpow2 82.2%

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

      2. *-un-lft-identity 82.2%

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

      3. times-frac 89.7%

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

   4. Applied egg-rr 89.7%

      \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. unpow2 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. times-frac 94.7%

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

   6. Applied egg-rr 94.7%

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

      1. /-rgt-identity 94.7%

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

      2. associate-*r/ 89.7%

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

      3. associate-/l* 94.7%

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

   8. Applied egg-rr 94.7%

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

   if 2.2000000000000001e158 < x

   1. Initial program 72.0%

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

      1. add-sqr-sqrt 72.0%

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

      2. times-frac 72.0%

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

   3. Applied egg-rr 91.5%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)}{y} \cdot \frac{\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)}{2}\\ \end{array} \]

Alternative 3: 52.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= (* z z) 2e-280)
     (* 0.5 y)
     (if (<= (* z z) 1e+38)
       t_0
       (if (<= (* z z) 1e+60)
         (* 0.5 y)
         (if (<= (* z z) 5e+115)
           (/ (* x (* x 0.5)) y)
           (if (<= (* z z) 1e+180)
             (* 0.5 y)
             (if (<= (* z z) 2e+190) t_0 (* -0.5 (* z (* z (/ 1.0 y))))))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 5e+115) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+190) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (z * (z * (1.0 / y)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    if ((z * z) <= 2d-280) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 1d+38) then
        tmp = t_0
    else if ((z * z) <= 1d+60) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 5d+115) then
        tmp = (x * (x * 0.5d0)) / y
    else if ((z * z) <= 1d+180) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 2d+190) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (z * (z * (1.0d0 / y)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 5e+115) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+190) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (z * (z * (1.0 / y)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if (z * z) <= 2e-280:
		tmp = 0.5 * y
	elif (z * z) <= 1e+38:
		tmp = t_0
	elif (z * z) <= 1e+60:
		tmp = 0.5 * y
	elif (z * z) <= 5e+115:
		tmp = (x * (x * 0.5)) / y
	elif (z * z) <= 1e+180:
		tmp = 0.5 * y
	elif (z * z) <= 2e+190:
		tmp = t_0
	else:
		tmp = -0.5 * (z * (z * (1.0 / y)))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (Float64(z * z) <= 2e-280)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 1e+38)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e+60)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 5e+115)
		tmp = Float64(Float64(x * Float64(x * 0.5)) / y);
	elseif (Float64(z * z) <= 1e+180)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 2e+190)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(z * Float64(z * Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if ((z * z) <= 2e-280)
		tmp = 0.5 * y;
	elseif ((z * z) <= 1e+38)
		tmp = t_0;
	elseif ((z * z) <= 1e+60)
		tmp = 0.5 * y;
	elseif ((z * z) <= 5e+115)
		tmp = (x * (x * 0.5)) / y;
	elseif ((z * z) <= 1e+180)
		tmp = 0.5 * y;
	elseif ((z * z) <= 2e+190)
		tmp = t_0;
	else
		tmp = -0.5 * (z * (z * (1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e-280], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+38], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e+60], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+115], N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+180], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+190], t$95$0, N[(-0.5 * N[(z * N[(z * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 1.9999999999999999e-280 or 9.99999999999999977e37 < (*.f64 z z) < 9.9999999999999995e59 or 5.00000000000000008e115 < (*.f64 z z) < 1e180

   1. Initial program 59.3%

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

   2. Taylor expanded in y around inf 60.3%

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

   if 1.9999999999999999e-280 < (*.f64 z z) < 9.99999999999999977e37 or 1e180 < (*.f64 z z) < 2.0000000000000001e190

   1. Initial program 78.2%

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

   2. Taylor expanded in x around inf 53.2%

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

      1. associate-*r/ 53.2%

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

      2. associate-/l* 53.2%

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

   4. Simplified 53.2%

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

      1. associate-/r/ 53.2%

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

      2. unpow2 53.2%

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

      3. associate-*r* 59.6%

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

   6. Applied egg-rr 59.6%

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

   if 9.9999999999999995e59 < (*.f64 z z) < 5.00000000000000008e115

   1. Initial program 87.4%

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

   2. Taylor expanded in x around inf 54.9%

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

      1. associate-*r/ 54.9%

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

      2. associate-/l* 54.9%

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

   4. Simplified 54.9%

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

      1. associate-/r/ 54.8%

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

      2. unpow2 54.8%

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

      3. associate-*r* 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l/ 54.7%

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

      3. associate-*r/ 54.9%

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

   8. Applied egg-rr 54.9%

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

   if 2.0000000000000001e190 < (*.f64 z z)

   1. Initial program 60.2%

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

   2. Taylor expanded in z around inf 71.9%

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

      1. *-commutative 71.9%

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

   4. Simplified 71.9%

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

      1. div-inv 71.9%

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

      2. unpow2 71.9%

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

      3. associate-*l* 77.9%

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

   6. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)} \cdot -0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{+38}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+190}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\\ \end{array} \]

Alternative 4: 50.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 3 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 4.5 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 6 \cdot 10^{+179}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1.85 \cdot 10^{+197}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= (* z z) 3e-280)
     (* 0.5 y)
     (if (<= (* z z) 4.5e+39)
       t_0
       (if (<= (* z z) 7.8e+59)
         (* 0.5 y)
         (if (<= (* z z) 9.5e+121)
           t_0
           (if (<= (* z z) 6e+179)
             (* 0.5 y)
             (if (<= (* z z) 1.85e+197) t_0 (* -0.5 (/ (* z z) y))))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 3e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 4.5e+39) {
		tmp = t_0;
	} else if ((z * z) <= 7.8e+59) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 9.5e+121) {
		tmp = t_0;
	} else if ((z * z) <= 6e+179) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1.85e+197) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((z * z) / y);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    if ((z * z) <= 3d-280) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 4.5d+39) then
        tmp = t_0
    else if ((z * z) <= 7.8d+59) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 9.5d+121) then
        tmp = t_0
    else if ((z * z) <= 6d+179) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 1.85d+197) then
        tmp = t_0
    else
        tmp = (-0.5d0) * ((z * z) / y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 3e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 4.5e+39) {
		tmp = t_0;
	} else if ((z * z) <= 7.8e+59) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 9.5e+121) {
		tmp = t_0;
	} else if ((z * z) <= 6e+179) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1.85e+197) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((z * z) / y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if (z * z) <= 3e-280:
		tmp = 0.5 * y
	elif (z * z) <= 4.5e+39:
		tmp = t_0
	elif (z * z) <= 7.8e+59:
		tmp = 0.5 * y
	elif (z * z) <= 9.5e+121:
		tmp = t_0
	elif (z * z) <= 6e+179:
		tmp = 0.5 * y
	elif (z * z) <= 1.85e+197:
		tmp = t_0
	else:
		tmp = -0.5 * ((z * z) / y)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (Float64(z * z) <= 3e-280)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 4.5e+39)
		tmp = t_0;
	elseif (Float64(z * z) <= 7.8e+59)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 9.5e+121)
		tmp = t_0;
	elseif (Float64(z * z) <= 6e+179)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 1.85e+197)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if ((z * z) <= 3e-280)
		tmp = 0.5 * y;
	elseif ((z * z) <= 4.5e+39)
		tmp = t_0;
	elseif ((z * z) <= 7.8e+59)
		tmp = 0.5 * y;
	elseif ((z * z) <= 9.5e+121)
		tmp = t_0;
	elseif ((z * z) <= 6e+179)
		tmp = 0.5 * y;
	elseif ((z * z) <= 1.85e+197)
		tmp = t_0;
	else
		tmp = -0.5 * ((z * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 3e-280], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4.5e+39], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 7.8e+59], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 9.5e+121], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 6e+179], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1.85e+197], t$95$0, N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.99999999999999987e-280 or 4.49999999999999996e39 < (*.f64 z z) < 7.80000000000000043e59 or 9.49999999999999949e121 < (*.f64 z z) < 5.9999999999999996e179

   1. Initial program 59.3%

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

   2. Taylor expanded in y around inf 60.3%

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

   if 2.99999999999999987e-280 < (*.f64 z z) < 4.49999999999999996e39 or 7.80000000000000043e59 < (*.f64 z z) < 9.49999999999999949e121 or 5.9999999999999996e179 < (*.f64 z z) < 1.8500000000000002e197

   1. Initial program 79.8%

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

   2. Taylor expanded in x around inf 53.5%

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

      1. associate-*r/ 53.5%

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

      2. associate-/l* 53.5%

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

   4. Simplified 53.5%

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

      1. associate-/r/ 53.5%

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

      2. unpow2 53.5%

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

      3. associate-*r* 58.8%

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

   6. Applied egg-rr 58.8%

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

   if 1.8500000000000002e197 < (*.f64 z z)

   1. Initial program 60.2%

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

   2. Taylor expanded in z around inf 71.9%

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

      1. *-commutative 71.9%

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

   4. Simplified 71.9%

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

      1. unpow2 71.9%

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

   6. Applied egg-rr 71.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 4.5 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 7.8 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 9.5 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 6 \cdot 10^{+179}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1.85 \cdot 10^{+197}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \]

Alternative 5: 50.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 2.2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1.32 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 7.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 9.8 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= (* z z) 2.2e-280)
     (* 0.5 y)
     (if (<= (* z z) 1.32e+38)
       t_0
       (if (<= (* z z) 1e+60)
         (* 0.5 y)
         (if (<= (* z z) 7.8e+116)
           (/ (* x (* x 0.5)) y)
           (if (<= (* z z) 1e+180)
             (* 0.5 y)
             (if (<= (* z z) 9.8e+191) t_0 (* -0.5 (/ (* z z) y))))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2.2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1.32e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 7.8e+116) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 9.8e+191) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((z * z) / y);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    if ((z * z) <= 2.2d-280) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 1.32d+38) then
        tmp = t_0
    else if ((z * z) <= 1d+60) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 7.8d+116) then
        tmp = (x * (x * 0.5d0)) / y
    else if ((z * z) <= 1d+180) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 9.8d+191) then
        tmp = t_0
    else
        tmp = (-0.5d0) * ((z * z) / y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2.2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1.32e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 7.8e+116) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 9.8e+191) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((z * z) / y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if (z * z) <= 2.2e-280:
		tmp = 0.5 * y
	elif (z * z) <= 1.32e+38:
		tmp = t_0
	elif (z * z) <= 1e+60:
		tmp = 0.5 * y
	elif (z * z) <= 7.8e+116:
		tmp = (x * (x * 0.5)) / y
	elif (z * z) <= 1e+180:
		tmp = 0.5 * y
	elif (z * z) <= 9.8e+191:
		tmp = t_0
	else:
		tmp = -0.5 * ((z * z) / y)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (Float64(z * z) <= 2.2e-280)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 1.32e+38)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e+60)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 7.8e+116)
		tmp = Float64(Float64(x * Float64(x * 0.5)) / y);
	elseif (Float64(z * z) <= 1e+180)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 9.8e+191)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if ((z * z) <= 2.2e-280)
		tmp = 0.5 * y;
	elseif ((z * z) <= 1.32e+38)
		tmp = t_0;
	elseif ((z * z) <= 1e+60)
		tmp = 0.5 * y;
	elseif ((z * z) <= 7.8e+116)
		tmp = (x * (x * 0.5)) / y;
	elseif ((z * z) <= 1e+180)
		tmp = 0.5 * y;
	elseif ((z * z) <= 9.8e+191)
		tmp = t_0;
	else
		tmp = -0.5 * ((z * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2.2e-280], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1.32e+38], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e+60], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 7.8e+116], N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+180], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 9.8e+191], t$95$0, N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 2.2000000000000001e-280 or 1.32e38 < (*.f64 z z) < 9.9999999999999995e59 or 7.80000000000000065e116 < (*.f64 z z) < 1e180

   1. Initial program 59.3%

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

   2. Taylor expanded in y around inf 60.3%

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

   if 2.2000000000000001e-280 < (*.f64 z z) < 1.32e38 or 1e180 < (*.f64 z z) < 9.7999999999999999e191

   1. Initial program 78.2%

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

   2. Taylor expanded in x around inf 53.2%

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

      1. associate-*r/ 53.2%

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

      2. associate-/l* 53.2%

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

   4. Simplified 53.2%

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

      1. associate-/r/ 53.2%

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

      2. unpow2 53.2%

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

      3. associate-*r* 59.6%

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

   6. Applied egg-rr 59.6%

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

   if 9.9999999999999995e59 < (*.f64 z z) < 7.80000000000000065e116

   1. Initial program 87.4%

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

   2. Taylor expanded in x around inf 54.9%

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

      1. associate-*r/ 54.9%

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

      2. associate-/l* 54.9%

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

   4. Simplified 54.9%

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

      1. associate-/r/ 54.8%

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

      2. unpow2 54.8%

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

      3. associate-*r* 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l/ 54.7%

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

      3. associate-*r/ 54.9%

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

   8. Applied egg-rr 54.9%

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

   if 9.7999999999999999e191 < (*.f64 z z)

   1. Initial program 60.2%

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

   2. Taylor expanded in z around inf 71.9%

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

      1. *-commutative 71.9%

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

   4. Simplified 71.9%

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

      1. unpow2 71.9%

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

   6. Applied egg-rr 71.9%

      \[\leadsto \frac{\color{blue}{z \cdot z}}{y} \cdot -0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1.32 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 7.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 9.8 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \]

Alternative 6: 52.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= (* z z) 2e-280)
     (* 0.5 y)
     (if (<= (* z z) 1e+38)
       t_0
       (if (<= (* z z) 1e+60)
         (* 0.5 y)
         (if (<= (* z z) 5e+115)
           (/ (* x (* x 0.5)) y)
           (if (<= (* z z) 1e+180)
             (* 0.5 y)
             (if (<= (* z z) 2e+190) t_0 (* (* z (/ z y)) -0.5)))))))))

C

x = abs(x);
double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 5e+115) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+190) {
		tmp = t_0;
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    if ((z * z) <= 2d-280) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 1d+38) then
        tmp = t_0
    else if ((z * z) <= 1d+60) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 5d+115) then
        tmp = (x * (x * 0.5d0)) / y
    else if ((z * z) <= 1d+180) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 2d+190) then
        tmp = t_0
    else
        tmp = (z * (z / y)) * (-0.5d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if ((z * z) <= 2e-280) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 1e+38) {
		tmp = t_0;
	} else if ((z * z) <= 1e+60) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 5e+115) {
		tmp = (x * (x * 0.5)) / y;
	} else if ((z * z) <= 1e+180) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+190) {
		tmp = t_0;
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if (z * z) <= 2e-280:
		tmp = 0.5 * y
	elif (z * z) <= 1e+38:
		tmp = t_0
	elif (z * z) <= 1e+60:
		tmp = 0.5 * y
	elif (z * z) <= 5e+115:
		tmp = (x * (x * 0.5)) / y
	elif (z * z) <= 1e+180:
		tmp = 0.5 * y
	elif (z * z) <= 2e+190:
		tmp = t_0
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (Float64(z * z) <= 2e-280)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 1e+38)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e+60)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 5e+115)
		tmp = Float64(Float64(x * Float64(x * 0.5)) / y);
	elseif (Float64(z * z) <= 1e+180)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 2e+190)
		tmp = t_0;
	else
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if ((z * z) <= 2e-280)
		tmp = 0.5 * y;
	elseif ((z * z) <= 1e+38)
		tmp = t_0;
	elseif ((z * z) <= 1e+60)
		tmp = 0.5 * y;
	elseif ((z * z) <= 5e+115)
		tmp = (x * (x * 0.5)) / y;
	elseif ((z * z) <= 1e+180)
		tmp = 0.5 * y;
	elseif ((z * z) <= 2e+190)
		tmp = t_0;
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e-280], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+38], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e+60], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+115], N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+180], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+190], t$95$0, N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 1.9999999999999999e-280 or 9.99999999999999977e37 < (*.f64 z z) < 9.9999999999999995e59 or 5.00000000000000008e115 < (*.f64 z z) < 1e180

   1. Initial program 59.3%

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

   2. Taylor expanded in y around inf 60.3%

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

   if 1.9999999999999999e-280 < (*.f64 z z) < 9.99999999999999977e37 or 1e180 < (*.f64 z z) < 2.0000000000000001e190

   1. Initial program 78.2%

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

   2. Taylor expanded in x around inf 53.2%

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

      1. associate-*r/ 53.2%

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

      2. associate-/l* 53.2%

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

   4. Simplified 53.2%

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

      1. associate-/r/ 53.2%

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

      2. unpow2 53.2%

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

      3. associate-*r* 59.6%

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

   6. Applied egg-rr 59.6%

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

   if 9.9999999999999995e59 < (*.f64 z z) < 5.00000000000000008e115

   1. Initial program 87.4%

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

   2. Taylor expanded in x around inf 54.9%

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

      1. associate-*r/ 54.9%

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

      2. associate-/l* 54.9%

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

   4. Simplified 54.9%

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

      1. associate-/r/ 54.8%

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

      2. unpow2 54.8%

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

      3. associate-*r* 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l/ 54.7%

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

      3. associate-*r/ 54.9%

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

   8. Applied egg-rr 54.9%

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

   if 2.0000000000000001e190 < (*.f64 z z)

   1. Initial program 60.2%

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

   2. Taylor expanded in z around inf 71.9%

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

      1. *-commutative 71.9%

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

   4. Simplified 71.9%

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

      1. unpow2 69.3%

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

      2. *-un-lft-identity 69.3%

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

      3. times-frac 81.8%

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

   6. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{y}\right)} \cdot -0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-280}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{+38}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+60}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+190}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 7: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+165} \lor \neg \left(y \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+165) (not (<= y 1.35e+154)))
   (* 0.5 y)
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+165) || !(y <= 1.35e+154)) {
		tmp = 0.5 * y;
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.05d+165)) .or. (.not. (y <= 1.35d+154))) then
        tmp = 0.5d0 * y
    else
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+165) || !(y <= 1.35e+154)) {
		tmp = 0.5 * y;
	} else {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if (y <= -1.05e+165) or not (y <= 1.35e+154):
		tmp = 0.5 * y
	else:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e+165) || !(y <= 1.35e+154))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.05e+165) || ~((y <= 1.35e+154)))
		tmp = 0.5 * y;
	else
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e+165], N[Not[LessEqual[y, 1.35e+154]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e165 or 1.35000000000000003e154 < y

   1. Initial program 9.8%

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

   2. Taylor expanded in y around inf 76.5%

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

   if -1.05e165 < y < 1.35000000000000003e154

   1. Initial program 86.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+165} \lor \neg \left(y \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 8: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+231}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+231)
   (+ (* 0.5 (- y (/ z (/ y z)))) (* 0.5 (* x (/ x y))))
   (* x (* x (/ 0.5 y)))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+231) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.05d+231) then
        tmp = (0.5d0 * (y - (z / (y / z)))) + (0.5d0 * (x * (x / y)))
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+231) {
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if x <= 1.05e+231:
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)))
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+231)
		tmp = Float64(Float64(0.5 * Float64(y - Float64(z / Float64(y / z)))) + Float64(0.5 * Float64(x * Float64(x / y))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.05e+231)
		tmp = (0.5 * (y - (z / (y / z)))) + (0.5 * (x * (x / y)));
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1.05e+231], N[(N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04999999999999992e231

   1. Initial program 65.3%

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

   2. Taylor expanded in x around inf 80.9%

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

      1. unpow2 80.9%

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

      2. *-un-lft-identity 80.9%

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

      3. times-frac 89.3%

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

   4. Applied egg-rr 89.3%

      \[\leadsto 0.5 \cdot \left(y - \frac{{z}^{2}}{y}\right) + 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{x}{y}\right)} \]
   5. Step-by-step derivation

      1. unpow2 89.3%

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

      2. *-un-lft-identity 89.3%

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

      3. times-frac 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. /-rgt-identity 94.0%

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

      2. associate-*r/ 89.3%

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

      3. associate-/l* 94.1%

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

   8. Applied egg-rr 94.1%

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

   if 1.04999999999999992e231 < x

   1. Initial program 86.7%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   4. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+231}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 9: 52.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 4.4e+84) (* 0.5 y) (* x (* x (/ 0.5 y)))))

C

x = abs(x);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.4e+84) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.4d+84) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.4e+84) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y, z):
	tmp = 0
	if x <= 4.4e+84:
		tmp = 0.5 * y
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

x = abs(x)
function code(x, y, z)
	tmp = 0.0
	if (x <= 4.4e+84)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.4e+84)
		tmp = 0.5 * y;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 4.4e+84], N[(0.5 * y), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3999999999999997e84

   1. Initial program 65.4%

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

   2. Taylor expanded in y around inf 40.0%

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

   if 4.3999999999999997e84 < x

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf 75.2%

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

      1. associate-*r/ 75.2%

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

      2. associate-/l* 75.2%

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

   4. Simplified 75.2%

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

      1. associate-/r/ 75.2%

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

      2. unpow2 75.2%

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

      3. associate-*r* 79.7%

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

   6. Applied egg-rr 79.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 10: 34.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0.5 \cdot y \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y z) :precision binary64 (* 0.5 y))

C

x = abs(x);
double code(double x, double y, double z) {
	return 0.5 * y;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * y
end function

Java

x = Math.abs(x);
public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

x = abs(x)
def code(x, y, z):
	return 0.5 * y

Julia

x = abs(x)
function code(x, y, z)
	return Float64(0.5 * y)
end

MATLAB

x = abs(x)
function tmp = code(x, y, z)
	tmp = 0.5 * y;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

2. Taylor expanded in y around inf 35.2%

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

3. Final simplification 35.2%

   \[\leadsto 0.5 \cdot y \]

Developer target: 99.9% 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 2023336 
(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)))