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

Percentage Accurate: 69.7% → 99.9%

Time: 7.6s

Alternatives: 9

Speedup: 1.4×

• 42.4% 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: 69.7% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 + (((z + x) / y) * (x - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

2. Taylor expanded in y around 0 82.2%

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

   1. distribute-lft-out 82.2%

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

   2. unpow2 82.2%

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

   3. unpow2 82.2%

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

   4. difference-of-squares 87.8%

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

   5. associate-/l* 99.9%

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

   6. +-commutative 99.9%

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

4. Simplified 99.9%

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

   1. associate-/r/ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-93}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ x (/ y x)))))
   (if (<= y -1200000000000.0)
     (* 0.5 y)
     (if (<= y 3.2e-129)
       t_0
       (if (<= y 8e-93)
         (* (* z (/ z y)) -0.5)
         (if (<= y 1.8e+25) t_0 (* 0.5 y)))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (y <= -1200000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 3.2e-129) {
		tmp = t_0;
	} else if (y <= 8e-93) {
		tmp = (z * (z / y)) * -0.5;
	} else if (y <= 1.8e+25) {
		tmp = t_0;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 = 0.5d0 * (x / (y / x))
    if (y <= (-1200000000000.0d0)) then
        tmp = 0.5d0 * y
    else if (y <= 3.2d-129) then
        tmp = t_0
    else if (y <= 8d-93) then
        tmp = (z * (z / y)) * (-0.5d0)
    else if (y <= 1.8d+25) then
        tmp = t_0
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x / (y / x));
	double tmp;
	if (y <= -1200000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 3.2e-129) {
		tmp = t_0;
	} else if (y <= 8e-93) {
		tmp = (z * (z / y)) * -0.5;
	} else if (y <= 1.8e+25) {
		tmp = t_0;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	t_0 = 0.5 * (x / (y / x))
	tmp = 0
	if y <= -1200000000000.0:
		tmp = 0.5 * y
	elif y <= 3.2e-129:
		tmp = t_0
	elif y <= 8e-93:
		tmp = (z * (z / y)) * -0.5
	elif y <= 1.8e+25:
		tmp = t_0
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x / Float64(y / x)))
	tmp = 0.0
	if (y <= -1200000000000.0)
		tmp = Float64(0.5 * y);
	elseif (y <= 3.2e-129)
		tmp = t_0;
	elseif (y <= 8e-93)
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	elseif (y <= 1.8e+25)
		tmp = t_0;
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x / (y / x));
	tmp = 0.0;
	if (y <= -1200000000000.0)
		tmp = 0.5 * y;
	elseif (y <= 3.2e-129)
		tmp = t_0;
	elseif (y <= 8e-93)
		tmp = (z * (z / y)) * -0.5;
	elseif (y <= 1.8e+25)
		tmp = t_0;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1200000000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, 3.2e-129], t$95$0, If[LessEqual[y, 8e-93], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 1.8e+25], t$95$0, N[(0.5 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e12 or 1.80000000000000008e25 < y

   1. Initial program 45.5%

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

   2. Taylor expanded in y around inf 68.8%

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

   if -1.2e12 < y < 3.2000000000000003e-129 or 7.9999999999999992e-93 < y < 1.80000000000000008e25

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf 54.6%

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

      1. unpow2 54.6%

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

      2. associate-/l* 56.2%

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

   4. Simplified 56.2%

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

   if 3.2000000000000003e-129 < y < 7.9999999999999992e-93

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in z around 0 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*r/ 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-93}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 3: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y -175000000000.0)
   (* 0.5 y)
   (if (<= y 9e-130)
     (* x (/ x (* y 2.0)))
     (if (<= y 9e-98)
       (* (* z (/ z y)) -0.5)
       (if (<= y 1.85e+25) (* 0.5 (/ x (/ y x))) (* 0.5 y))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -175000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 9e-130) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 9e-98) {
		tmp = (z * (z / y)) * -0.5;
	} else if (y <= 1.85e+25) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 <= (-175000000000.0d0)) then
        tmp = 0.5d0 * y
    else if (y <= 9d-130) then
        tmp = x * (x / (y * 2.0d0))
    else if (y <= 9d-98) then
        tmp = (z * (z / y)) * (-0.5d0)
    else if (y <= 1.85d+25) then
        tmp = 0.5d0 * (x / (y / x))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -175000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 9e-130) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 9e-98) {
		tmp = (z * (z / y)) * -0.5;
	} else if (y <= 1.85e+25) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if y <= -175000000000.0:
		tmp = 0.5 * y
	elif y <= 9e-130:
		tmp = x * (x / (y * 2.0))
	elif y <= 9e-98:
		tmp = (z * (z / y)) * -0.5
	elif y <= 1.85e+25:
		tmp = 0.5 * (x / (y / x))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (y <= -175000000000.0)
		tmp = Float64(0.5 * y);
	elseif (y <= 9e-130)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (y <= 9e-98)
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	elseif (y <= 1.85e+25)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -175000000000.0)
		tmp = 0.5 * y;
	elseif (y <= 9e-130)
		tmp = x * (x / (y * 2.0));
	elseif (y <= 9e-98)
		tmp = (z * (z / y)) * -0.5;
	elseif (y <= 1.85e+25)
		tmp = 0.5 * (x / (y / x));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, -175000000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, 9e-130], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-98], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 1.85e+25], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.75e11 or 1.8499999999999999e25 < y

   1. Initial program 45.5%

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

   2. Taylor expanded in y around inf 68.8%

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

   if -1.75e11 < y < 9e-130

   1. Initial program 88.9%

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

   2. Taylor expanded in x around inf 52.7%

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

      1. unpow2 52.7%

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

   4. Simplified 52.7%

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

      1. associate-/l* 54.6%

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

      2. associate-/r/ 54.6%

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

   6. Applied egg-rr 54.6%

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

   if 9e-130 < y < 8.99999999999999994e-98

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

   4. Simplified 83.2%

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

   5. Taylor expanded in z around 0 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*r/ 83.2%

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

   7. Simplified 83.2%

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

   if 8.99999999999999994e-98 < y < 1.8499999999999999e25

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. unpow2 64.9%

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

      2. associate-/l* 64.9%

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

   4. Simplified 64.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 4: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -56000000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-95}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y -56000000000000.0)
   (* 0.5 y)
   (if (<= y 1.7e-128)
     (* x (/ x (* y 2.0)))
     (if (<= y 1.15e-95)
       (* -0.5 (/ z (/ y z)))
       (if (<= y 1.65e+25) (* 0.5 (/ x (/ y x))) (* 0.5 y))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -56000000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 1.7e-128) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 1.15e-95) {
		tmp = -0.5 * (z / (y / z));
	} else if (y <= 1.65e+25) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 <= (-56000000000000.0d0)) then
        tmp = 0.5d0 * y
    else if (y <= 1.7d-128) then
        tmp = x * (x / (y * 2.0d0))
    else if (y <= 1.15d-95) then
        tmp = (-0.5d0) * (z / (y / z))
    else if (y <= 1.65d+25) then
        tmp = 0.5d0 * (x / (y / x))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -56000000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 1.7e-128) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 1.15e-95) {
		tmp = -0.5 * (z / (y / z));
	} else if (y <= 1.65e+25) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if y <= -56000000000000.0:
		tmp = 0.5 * y
	elif y <= 1.7e-128:
		tmp = x * (x / (y * 2.0))
	elif y <= 1.15e-95:
		tmp = -0.5 * (z / (y / z))
	elif y <= 1.65e+25:
		tmp = 0.5 * (x / (y / x))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (y <= -56000000000000.0)
		tmp = Float64(0.5 * y);
	elseif (y <= 1.7e-128)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (y <= 1.15e-95)
		tmp = Float64(-0.5 * Float64(z / Float64(y / z)));
	elseif (y <= 1.65e+25)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -56000000000000.0)
		tmp = 0.5 * y;
	elseif (y <= 1.7e-128)
		tmp = x * (x / (y * 2.0));
	elseif (y <= 1.15e-95)
		tmp = -0.5 * (z / (y / z));
	elseif (y <= 1.65e+25)
		tmp = 0.5 * (x / (y / x));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, -56000000000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, 1.7e-128], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e-95], N[(-0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+25], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.6e13 or 1.6500000000000001e25 < y

   1. Initial program 45.5%

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

   2. Taylor expanded in y around inf 68.8%

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

   if -5.6e13 < y < 1.69999999999999987e-128

   1. Initial program 88.9%

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

   2. Taylor expanded in x around inf 52.7%

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

      1. unpow2 52.7%

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

   4. Simplified 52.7%

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

      1. associate-/l* 54.6%

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

      2. associate-/r/ 54.6%

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

   6. Applied egg-rr 54.6%

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

   if 1.69999999999999987e-128 < y < 1.15e-95

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. associate-/l* 83.5%

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

   4. Simplified 83.5%

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

   if 1.15e-95 < y < 1.6500000000000001e25

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. unpow2 64.9%

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

      2. associate-/l* 64.9%

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

   4. Simplified 64.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -56000000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-95}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 5: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{y + z}{y} \cdot \left(y - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+196)
   (* 0.5 (+ y (* x (/ x y))))
   (* 0.5 (* (/ (+ y z) y) (- y z)))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+196) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (((y + z) / y) * (y - z));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 ((z * z) <= 2d+196) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = 0.5d0 * (((y + z) / y) * (y - z))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+196) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (((y + z) / y) * (y - z));
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+196:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = 0.5 * (((y + z) / y) * (y - z))
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+196)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(y + z) / y) * Float64(y - z)));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+196)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = 0.5 * (((y + z) / y) * (y - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+196], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(y + z), $MachinePrecision] / y), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e196

   1. Initial program 73.3%

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

   2. Taylor expanded in y around 0 91.6%

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

      1. distribute-lft-out 91.6%

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

      2. unpow2 91.6%

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

      3. unpow2 91.6%

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

      4. difference-of-squares 91.6%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.0%

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

      1. unpow2 81.0%

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

      2. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. associate-/r/ 88.7%

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

   9. Applied egg-rr 88.7%

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

   if 1.9999999999999999e196 < (*.f64 z z)

   1. Initial program 51.5%

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

   2. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. difference-of-squares 62.1%

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

      4. associate-/l* 88.3%

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

      5. associate-/r/ 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{y + z}{y} \cdot \left(y - z\right)\right)\\ \end{array} \]

Alternative 6: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+196)
   (* 0.5 (+ y (* x (/ x y))))
   (* 0.5 (- y (* z (/ z y))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+196) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 ((z * z) <= 2d+196) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+196) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+196:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+196)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+196)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+196], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e196

   1. Initial program 73.3%

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

   2. Taylor expanded in y around 0 91.6%

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

      1. distribute-lft-out 91.6%

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

      2. unpow2 91.6%

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

      3. unpow2 91.6%

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

      4. difference-of-squares 91.6%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.0%

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

      1. unpow2 81.0%

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

      2. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. associate-/r/ 88.7%

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

   9. Applied egg-rr 88.7%

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

   if 1.9999999999999999e196 < (*.f64 z z)

   1. Initial program 51.5%

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

   2. Taylor expanded in y around 0 60.3%

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

      1. distribute-lft-out 60.3%

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

      2. unpow2 60.3%

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

      3. unpow2 60.3%

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

      4. difference-of-squares 78.8%

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

      5. associate-/l* 100.0%

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

      6. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. sub-neg 70.6%

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

      3. unpow2 70.6%

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

      4. associate-*r/ 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+196}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 7: 51.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -30000000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y -30000000000000.0)
   (* 0.5 y)
   (if (<= y 4.8e+24) (* 0.5 (/ x (/ y x))) (* 0.5 y))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -30000000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 4.8e+24) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 <= (-30000000000000.0d0)) then
        tmp = 0.5d0 * y
    else if (y <= 4.8d+24) then
        tmp = 0.5d0 * (x / (y / x))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -30000000000000.0) {
		tmp = 0.5 * y;
	} else if (y <= 4.8e+24) {
		tmp = 0.5 * (x / (y / x));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if y <= -30000000000000.0:
		tmp = 0.5 * y
	elif y <= 4.8e+24:
		tmp = 0.5 * (x / (y / x))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (y <= -30000000000000.0)
		tmp = Float64(0.5 * y);
	elseif (y <= 4.8e+24)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -30000000000000.0)
		tmp = 0.5 * y;
	elseif (y <= 4.8e+24)
		tmp = 0.5 * (x / (y / x));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, -30000000000000.0], N[(0.5 * y), $MachinePrecision], If[LessEqual[y, 4.8e+24], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e13 or 4.8000000000000001e24 < y

   1. Initial program 45.5%

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

   2. Taylor expanded in y around inf 68.8%

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

   if -3e13 < y < 4.8000000000000001e24

   1. Initial program 90.1%

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

   2. Taylor expanded in x around inf 52.4%

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

      1. unpow2 52.4%

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

      2. associate-/l* 54.0%

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

   4. Simplified 54.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000000000000:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 8: 78.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z 2.4e+194) (* 0.5 (+ y (* x (/ x y)))) (* (* z (/ z y)) -0.5)))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+194) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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 (z <= 2.4d+194) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = (z * (z / y)) * (-0.5d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+194) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if z <= 2.4e+194:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e+194)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.4e+194)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, 2.4e+194], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e194

   1. Initial program 68.1%

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

   2. Taylor expanded in y around 0 85.0%

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

      1. distribute-lft-out 85.0%

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

      2. unpow2 85.0%

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

      3. unpow2 85.0%

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

      4. difference-of-squares 88.5%

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

      5. associate-/l* 99.9%

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

      6. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 71.3%

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

      1. unpow2 71.3%

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

      2. associate-/l* 78.6%

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

   7. Simplified 78.6%

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

      1. associate-/r/ 78.6%

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

   9. Applied egg-rr 78.6%

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

   if 2.4e194 < z

   1. Initial program 54.2%

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

   2. Taylor expanded in z around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. unpow2 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in z around 0 76.2%

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

      1. unpow2 76.2%

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

      2. associate-*r/ 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+194}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 9: 33.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: z 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);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

2. Taylor expanded in y around inf 39.4%

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

3. Final simplification 39.4%

   \[\leadsto 0.5 \cdot y \]

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 2023290 
(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)))