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

Percentage Accurate: 69.2% → 99.9%

Time: 11.8s

Alternatives: 10

Speedup: 1.2×

• 42.0% 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: 69.2% 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

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

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

Wolfram

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 71.8%

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

2. Taylor expanded in y around 0 85.2%

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

   1. +-lft-identity 85.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   2. +-commutative 85.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   3. mul0-lft 85.2%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

   4. metadata-eval 85.2%

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

   5. distribute-rgt1-in 85.2%

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

   6. distribute-lft-out 85.2%

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

   7. distribute-rgt1-in 85.2%

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

   8. metadata-eval 85.2%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   9. mul0-lft 85.2%

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

   10. +-rgt-identity 85.2%

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

   11. unpow2 85.2%

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

   12. unpow2 85.2%

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

   13. difference-of-squares 90.8%

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

   14. associate-/l* 99.9%

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

   15. +-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: 52.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e-62)
   (* 0.5 (* x (/ x y)))
   (if (<= (* z z) 4e-35)
     (* 0.5 y)
     (if (<= (* z z) 2e+30)
       (* 0.5 (/ x (/ y x)))
       (if (<= (* z z) 2e+113)
         (* (* z z) (/ -0.5 y))
         (if (<= (* z z) 5e+190) (* 0.5 y) (* (* z (/ z y)) -0.5)))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-62) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 4e-35) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+30) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z * z) <= 2e+113) {
		tmp = (z * z) * (-0.5 / y);
	} else if ((z * z) <= 5e+190) {
		tmp = 0.5 * y;
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Fortran

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) <= 4d-62) then
        tmp = 0.5d0 * (x * (x / y))
    else if ((z * z) <= 4d-35) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 2d+30) then
        tmp = 0.5d0 * (x / (y / x))
    else if ((z * z) <= 2d+113) then
        tmp = (z * z) * ((-0.5d0) / y)
    else if ((z * z) <= 5d+190) then
        tmp = 0.5d0 * y
    else
        tmp = (z * (z / y)) * (-0.5d0)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-62) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 4e-35) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+30) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z * z) <= 2e+113) {
		tmp = (z * z) * (-0.5 / y);
	} else if ((z * z) <= 5e+190) {
		tmp = 0.5 * y;
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e-62:
		tmp = 0.5 * (x * (x / y))
	elif (z * z) <= 4e-35:
		tmp = 0.5 * y
	elif (z * z) <= 2e+30:
		tmp = 0.5 * (x / (y / x))
	elif (z * z) <= 2e+113:
		tmp = (z * z) * (-0.5 / y)
	elif (z * z) <= 5e+190:
		tmp = 0.5 * y
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e-62)
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	elseif (Float64(z * z) <= 4e-35)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 2e+30)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	elseif (Float64(z * z) <= 2e+113)
		tmp = Float64(Float64(z * z) * Float64(-0.5 / y));
	elseif (Float64(z * z) <= 5e+190)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e-62)
		tmp = 0.5 * (x * (x / y));
	elseif ((z * z) <= 4e-35)
		tmp = 0.5 * y;
	elseif ((z * z) <= 2e+30)
		tmp = 0.5 * (x / (y / x));
	elseif ((z * z) <= 2e+113)
		tmp = (z * z) * (-0.5 / y);
	elseif ((z * z) <= 5e+190)
		tmp = 0.5 * y;
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e-62], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e-35], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+30], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+113], N[(N[(z * z), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+190], N[(0.5 * y), $MachinePrecision], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z z) < 4.0000000000000002e-62

   1. Initial program 77.4%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/l* 60.1%

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

   4. Simplified 60.1%

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

      1. associate-/r/ 60.1%

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

   6. Applied egg-rr 60.1%

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

   if 4.0000000000000002e-62 < (*.f64 z z) < 4.00000000000000003e-35 or 2e113 < (*.f64 z z) < 5.00000000000000036e190

   1. Initial program 78.8%

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

   2. Taylor expanded in y around inf 59.7%

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

   if 4.00000000000000003e-35 < (*.f64 z z) < 2e30

   1. Initial program 78.9%

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

   2. Taylor expanded in x around inf 48.7%

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

      1. unpow2 48.7%

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

      2. associate-/l* 62.6%

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

   4. Simplified 62.6%

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

   if 2e30 < (*.f64 z z) < 2e113

   1. Initial program 89.0%

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

   2. Taylor expanded in z around inf 66.5%

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

      1. *-commutative 66.5%

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

      2. unpow2 66.5%

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

      3. associate-/l* 66.5%

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

   4. Simplified 66.5%

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

      1. *-commutative 66.5%

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

      2. clear-num 66.7%

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

      3. un-div-inv 66.7%

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

      4. associate-/l/ 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. associate-/r/ 66.7%

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

   8. Applied egg-rr 66.7%

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

   if 5.00000000000000036e190 < (*.f64 z z)

   1. Initial program 61.7%

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

   2. Taylor expanded in z around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. associate-*l/ 70.9%

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

      4. *-commutative 70.9%

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

   4. Simplified 70.9%

      \[\leadsto \color{blue}{\left(z \cdot \frac{z}{y}\right) \cdot -0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 3: 52.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e-62)
   (* 0.5 (* x (/ x y)))
   (if (<= (* z z) 4e-35)
     (* 0.5 y)
     (if (<= (* z z) 2e+30)
       (* 0.5 (/ x (/ y x)))
       (if (<= (* z z) 2e+113)
         (* (* z z) (/ -0.5 y))
         (if (<= (* z z) 5e+190) (* 0.5 y) (* (/ z (/ y z)) -0.5)))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-62) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 4e-35) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+30) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z * z) <= 2e+113) {
		tmp = (z * z) * (-0.5 / y);
	} else if ((z * z) <= 5e+190) {
		tmp = 0.5 * y;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Fortran

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) <= 4d-62) then
        tmp = 0.5d0 * (x * (x / y))
    else if ((z * z) <= 4d-35) then
        tmp = 0.5d0 * y
    else if ((z * z) <= 2d+30) then
        tmp = 0.5d0 * (x / (y / x))
    else if ((z * z) <= 2d+113) then
        tmp = (z * z) * ((-0.5d0) / y)
    else if ((z * z) <= 5d+190) then
        tmp = 0.5d0 * y
    else
        tmp = (z / (y / z)) * (-0.5d0)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-62) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 4e-35) {
		tmp = 0.5 * y;
	} else if ((z * z) <= 2e+30) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z * z) <= 2e+113) {
		tmp = (z * z) * (-0.5 / y);
	} else if ((z * z) <= 5e+190) {
		tmp = 0.5 * y;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e-62:
		tmp = 0.5 * (x * (x / y))
	elif (z * z) <= 4e-35:
		tmp = 0.5 * y
	elif (z * z) <= 2e+30:
		tmp = 0.5 * (x / (y / x))
	elif (z * z) <= 2e+113:
		tmp = (z * z) * (-0.5 / y)
	elif (z * z) <= 5e+190:
		tmp = 0.5 * y
	else:
		tmp = (z / (y / z)) * -0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e-62)
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	elseif (Float64(z * z) <= 4e-35)
		tmp = Float64(0.5 * y);
	elseif (Float64(z * z) <= 2e+30)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	elseif (Float64(z * z) <= 2e+113)
		tmp = Float64(Float64(z * z) * Float64(-0.5 / y));
	elseif (Float64(z * z) <= 5e+190)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(z / Float64(y / z)) * -0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e-62)
		tmp = 0.5 * (x * (x / y));
	elseif ((z * z) <= 4e-35)
		tmp = 0.5 * y;
	elseif ((z * z) <= 2e+30)
		tmp = 0.5 * (x / (y / x));
	elseif ((z * z) <= 2e+113)
		tmp = (z * z) * (-0.5 / y);
	elseif ((z * z) <= 5e+190)
		tmp = 0.5 * y;
	else
		tmp = (z / (y / z)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e-62], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e-35], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+30], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+113], N[(N[(z * z), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+190], N[(0.5 * y), $MachinePrecision], N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z z) < 4.0000000000000002e-62

   1. Initial program 77.4%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/l* 60.1%

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

   4. Simplified 60.1%

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

      1. associate-/r/ 60.1%

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

   6. Applied egg-rr 60.1%

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

   if 4.0000000000000002e-62 < (*.f64 z z) < 4.00000000000000003e-35 or 2e113 < (*.f64 z z) < 5.00000000000000036e190

   1. Initial program 78.8%

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

   2. Taylor expanded in y around inf 59.7%

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

   if 4.00000000000000003e-35 < (*.f64 z z) < 2e30

   1. Initial program 78.9%

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

   2. Taylor expanded in x around inf 48.7%

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

      1. unpow2 48.7%

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

      2. associate-/l* 62.6%

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

   4. Simplified 62.6%

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

   if 2e30 < (*.f64 z z) < 2e113

   1. Initial program 89.0%

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

   2. Taylor expanded in z around inf 66.5%

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

      1. *-commutative 66.5%

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

      2. unpow2 66.5%

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

      3. associate-/l* 66.5%

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

   4. Simplified 66.5%

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

      1. *-commutative 66.5%

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

      2. clear-num 66.7%

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

      3. un-div-inv 66.7%

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

      4. associate-/l/ 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. associate-/r/ 66.7%

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

   8. Applied egg-rr 66.7%

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

   if 5.00000000000000036e190 < (*.f64 z z)

   1. Initial program 61.7%

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

   2. Taylor expanded in z around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. associate-/l* 70.9%

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

   4. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{z}{\frac{y}{z}} \cdot -0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-62}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \]

Alternative 4: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+157} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+221}\right):\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{z + x}{y} \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+30)
   (* 0.5 (+ y (* x (/ x y))))
   (if (or (<= (* z z) 5e+157) (not (<= (* z z) 2e+221)))
     (* 0.5 (- y (/ z (/ y z))))
     (* 0.5 (* (/ (+ z x) y) (- x z))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+30) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else if (((z * z) <= 5e+157) || !((z * z) <= 2e+221)) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (((z + x) / y) * (x - z));
	}
	return tmp;
}

Fortran

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+30) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else if (((z * z) <= 5d+157) .or. (.not. ((z * z) <= 2d+221))) then
        tmp = 0.5d0 * (y - (z / (y / z)))
    else
        tmp = 0.5d0 * (((z + x) / y) * (x - z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+30) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else if (((z * z) <= 5e+157) || !((z * z) <= 2e+221)) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else {
		tmp = 0.5 * (((z + x) / y) * (x - z));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+30:
		tmp = 0.5 * (y + (x * (x / y)))
	elif ((z * z) <= 5e+157) or not ((z * z) <= 2e+221):
		tmp = 0.5 * (y - (z / (y / z)))
	else:
		tmp = 0.5 * (((z + x) / y) * (x - z))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+30)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	elseif ((Float64(z * z) <= 5e+157) || !(Float64(z * z) <= 2e+221))
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(z + x) / y) * Float64(x - z)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+30)
		tmp = 0.5 * (y + (x * (x / y)));
	elseif (((z * z) <= 5e+157) || ~(((z * z) <= 2e+221)))
		tmp = 0.5 * (y - (z / (y / z)));
	else
		tmp = 0.5 * (((z + x) / y) * (x - z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+30], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(z * z), $MachinePrecision], 5e+157], N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e+221]], $MachinePrecision]], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e30

   1. Initial program 77.9%

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

   2. Taylor expanded in y around 0 93.3%

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

      1. +-lft-identity 93.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 93.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 93.3%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 93.3%

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

      5. distribute-rgt1-in 93.3%

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

      6. distribute-lft-out 93.3%

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

      7. distribute-rgt1-in 93.3%

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

      8. metadata-eval 93.3%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 93.3%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 93.3%

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

      11. unpow2 93.3%

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

      12. unpow2 93.3%

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

      13. difference-of-squares 93.3%

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

      14. associate-/l* 99.9%

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

      15. +-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 89.1%

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

      1. unpow2 89.1%

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

      2. associate-*r/ 95.7%

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

   7. Simplified 95.7%

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

   if 2e30 < (*.f64 z z) < 4.99999999999999976e157 or 2.0000000000000001e221 < (*.f64 z z)

   1. Initial program 61.6%

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

   2. Taylor expanded in x around 0 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. div-sub 62.8%

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

      4. associate-/l* 75.5%

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

      5. *-inverses 75.5%

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

      6. /-rgt-identity 75.5%

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

      7. associate-/l* 89.1%

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

   4. Simplified 89.1%

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

   if 4.99999999999999976e157 < (*.f64 z z) < 2.0000000000000001e221

   1. Initial program 92.8%

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

   2. Taylor expanded in y around 0 84.8%

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

      1. unpow2 84.8%

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

      2. unpow2 84.8%

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

      3. sub-neg 84.8%

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

      4. mul-1-neg 84.8%

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

      5. unpow2 84.8%

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

      6. +-commutative 84.8%

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

      7. unpow2 84.8%

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

      8. unpow2 84.8%

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

      9. +-commutative 84.8%

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

      10. unpow2 84.8%

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

      11. mul-1-neg 84.8%

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

      12. sub-neg 84.8%

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

      13. difference-of-squares 84.8%

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

      14. associate-/l* 84.8%

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

      15. +-commutative 84.8%

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

   4. Simplified 84.8%

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

      1. associate-/r/ 100.0%

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

   6. Applied egg-rr 84.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+157} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+221}\right):\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{z + x}{y} \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5: 51.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{if}\;z \leq 1.75 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+57} \lor \neg \left(z \leq 1.85 \cdot 10^{+95}\right):\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x (/ x y)))))
   (if (<= z 1.75e-253)
     t_0
     (if (<= z 3.1e-220)
       (* 0.5 y)
       (if (<= z 3.2e-31)
         t_0
         (if (<= z 6.6e-18)
           (* 0.5 y)
           (if (<= z 7.2e+21)
             (* 0.5 (/ x (/ y x)))
             (if (or (<= z 1.22e+57) (not (<= z 1.85e+95)))
               (* (* z (/ z y)) -0.5)
               (* 0.5 y)))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * (x / y));
	double tmp;
	if (z <= 1.75e-253) {
		tmp = t_0;
	} else if (z <= 3.1e-220) {
		tmp = 0.5 * y;
	} else if (z <= 3.2e-31) {
		tmp = t_0;
	} else if (z <= 6.6e-18) {
		tmp = 0.5 * y;
	} else if (z <= 7.2e+21) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z <= 1.22e+57) || !(z <= 1.85e+95)) {
		tmp = (z * (z / y)) * -0.5;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

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 * (x / y))
    if (z <= 1.75d-253) then
        tmp = t_0
    else if (z <= 3.1d-220) then
        tmp = 0.5d0 * y
    else if (z <= 3.2d-31) then
        tmp = t_0
    else if (z <= 6.6d-18) then
        tmp = 0.5d0 * y
    else if (z <= 7.2d+21) then
        tmp = 0.5d0 * (x / (y / x))
    else if ((z <= 1.22d+57) .or. (.not. (z <= 1.85d+95))) then
        tmp = (z * (z / y)) * (-0.5d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * (x / y));
	double tmp;
	if (z <= 1.75e-253) {
		tmp = t_0;
	} else if (z <= 3.1e-220) {
		tmp = 0.5 * y;
	} else if (z <= 3.2e-31) {
		tmp = t_0;
	} else if (z <= 6.6e-18) {
		tmp = 0.5 * y;
	} else if (z <= 7.2e+21) {
		tmp = 0.5 * (x / (y / x));
	} else if ((z <= 1.22e+57) || !(z <= 1.85e+95)) {
		tmp = (z * (z / y)) * -0.5;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = 0.5 * (x * (x / y))
	tmp = 0
	if z <= 1.75e-253:
		tmp = t_0
	elif z <= 3.1e-220:
		tmp = 0.5 * y
	elif z <= 3.2e-31:
		tmp = t_0
	elif z <= 6.6e-18:
		tmp = 0.5 * y
	elif z <= 7.2e+21:
		tmp = 0.5 * (x / (y / x))
	elif (z <= 1.22e+57) or not (z <= 1.85e+95):
		tmp = (z * (z / y)) * -0.5
	else:
		tmp = 0.5 * y
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x * Float64(x / y)))
	tmp = 0.0
	if (z <= 1.75e-253)
		tmp = t_0;
	elseif (z <= 3.1e-220)
		tmp = Float64(0.5 * y);
	elseif (z <= 3.2e-31)
		tmp = t_0;
	elseif (z <= 6.6e-18)
		tmp = Float64(0.5 * y);
	elseif (z <= 7.2e+21)
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	elseif ((z <= 1.22e+57) || !(z <= 1.85e+95))
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x * (x / y));
	tmp = 0.0;
	if (z <= 1.75e-253)
		tmp = t_0;
	elseif (z <= 3.1e-220)
		tmp = 0.5 * y;
	elseif (z <= 3.2e-31)
		tmp = t_0;
	elseif (z <= 6.6e-18)
		tmp = 0.5 * y;
	elseif (z <= 7.2e+21)
		tmp = 0.5 * (x / (y / x));
	elseif ((z <= 1.22e+57) || ~((z <= 1.85e+95)))
		tmp = (z * (z / y)) * -0.5;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.75e-253], t$95$0, If[LessEqual[z, 3.1e-220], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 3.2e-31], t$95$0, If[LessEqual[z, 6.6e-18], N[(0.5 * y), $MachinePrecision], If[LessEqual[z, 7.2e+21], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 1.22e+57], N[Not[LessEqual[z, 1.85e+95]], $MachinePrecision]], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.75000000000000011e-253 or 3.10000000000000011e-220 < z < 3.20000000000000018e-31

   1. Initial program 75.5%

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

   2. Taylor expanded in x around inf 40.4%

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

      1. unpow2 40.4%

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

      2. associate-/l* 44.0%

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

   4. Simplified 44.0%

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

      1. associate-/r/ 44.0%

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

   6. Applied egg-rr 44.0%

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

   if 1.75000000000000011e-253 < z < 3.10000000000000011e-220 or 3.20000000000000018e-31 < z < 6.6000000000000003e-18 or 1.22e57 < z < 1.8500000000000001e95

   1. Initial program 64.2%

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

   2. Taylor expanded in y around inf 62.0%

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

   if 6.6000000000000003e-18 < z < 7.2e21

   1. Initial program 69.3%

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

   2. Taylor expanded in x around inf 36.0%

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

      1. unpow2 36.0%

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

      2. associate-/l* 51.2%

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

   4. Simplified 51.2%

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

   if 7.2e21 < z < 1.22e57 or 1.8500000000000001e95 < z

   1. Initial program 62.9%

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

   2. Taylor expanded in z around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. associate-*l/ 64.3%

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

      4. *-commutative 64.3%

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

   4. Simplified 64.3%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+57} \lor \neg \left(z \leq 1.85 \cdot 10^{+95}\right):\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 6: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+30} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+113}\right) \land z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 2e+30)
         (and (not (<= (* z z) 2e+113)) (<= (* z z) 5e+190)))
   (* 0.5 (+ y (* x (/ x y))))
   (* 0.5 (- y (/ z (/ y z))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2e+30) || (!((z * z) <= 2e+113) && ((z * z) <= 5e+190))) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Fortran

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+30) .or. (.not. ((z * z) <= 2d+113)) .and. ((z * z) <= 5d+190)) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = 0.5d0 * (y - (z / (y / z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2e+30) || (!((z * z) <= 2e+113) && ((z * z) <= 5e+190))) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = 0.5 * (y - (z / (y / z)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if ((z * z) <= 2e+30) or (not ((z * z) <= 2e+113) and ((z * z) <= 5e+190)):
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = 0.5 * (y - (z / (y / z)))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 2e+30) || (!(Float64(z * z) <= 2e+113) && (Float64(z * z) <= 5e+190)))
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z * z) <= 2e+30) || (~(((z * z) <= 2e+113)) && ((z * z) <= 5e+190)))
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = 0.5 * (y - (z / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 2e+30], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e+113]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 5e+190]]], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e30 or 2e113 < (*.f64 z z) < 5.00000000000000036e190

   1. Initial program 77.7%

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

   2. Taylor expanded in y around 0 93.8%

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

      1. +-lft-identity 93.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 93.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 93.8%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 93.8%

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

      5. distribute-rgt1-in 93.8%

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

      6. distribute-lft-out 93.8%

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

      7. distribute-rgt1-in 93.8%

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

      8. metadata-eval 93.8%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 93.8%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 93.8%

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

      11. unpow2 93.8%

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

      12. unpow2 93.8%

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

      13. difference-of-squares 93.8%

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

      14. associate-/l* 99.9%

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

      15. +-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 89.3%

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

      1. unpow2 89.3%

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

      2. associate-*r/ 95.3%

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

   7. Simplified 95.3%

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

   if 2e30 < (*.f64 z z) < 2e113 or 5.00000000000000036e190 < (*.f64 z z)

   1. Initial program 63.9%

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

   2. Taylor expanded in x around 0 63.3%

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

      1. unpow2 63.3%

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

      2. unpow2 63.3%

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

      3. div-sub 63.3%

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

      4. associate-/l* 74.1%

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

      5. *-inverses 74.1%

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

      6. /-rgt-identity 74.1%

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

      7. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+30} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+113}\right) \land z \cdot z \leq 5 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 7: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+246) (* 0.5 (+ y (* x (/ x y)))) (* (* z (/ z y)) -0.5)))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+246) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Fortran

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) <= 5d+246) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = (z * (z / y)) * (-0.5d0)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+246) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = (z * (z / y)) * -0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+246:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = (z * (z / y)) * -0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+246)
		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

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+246)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = (z * (z / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+246], 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 (*.f64 z z) < 4.99999999999999976e246

   1. Initial program 78.4%

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

   2. Taylor expanded in y around 0 94.7%

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

      1. +-lft-identity 94.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(0 + y\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      2. +-commutative 94.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + 0\right)} + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      3. mul0-lft 94.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{0 \cdot z}\right) + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]

      4. metadata-eval 94.7%

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

      5. distribute-rgt1-in 94.7%

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

      6. distribute-lft-out 94.7%

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

      7. distribute-rgt1-in 94.7%

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

      8. metadata-eval 94.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0} \cdot z\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      9. mul0-lft 94.7%

         \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{0}\right) + \frac{{x}^{2} - {z}^{2}}{y}\right) \]

      10. +-rgt-identity 94.7%

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

      11. unpow2 94.7%

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

      12. unpow2 94.7%

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

      13. difference-of-squares 94.7%

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

      14. associate-/l* 99.9%

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

      15. +-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 82.9%

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

      1. unpow2 82.9%

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

      2. associate-*r/ 88.0%

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

   7. Simplified 88.0%

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

   if 4.99999999999999976e246 < (*.f64 z z)

   1. Initial program 57.9%

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

   2. Taylor expanded in z around inf 71.8%

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

      1. *-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. associate-*l/ 76.6%

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

      4. *-commutative 76.6%

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

   4. Simplified 76.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+246}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]

Alternative 8: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.95e-16) (* 0.5 y) (* 0.5 (* x (/ x y)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.95e-16) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Fortran

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 (x <= 1.95d-16) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (x * (x / y))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.95e-16) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (x / y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 1.95e-16:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x * (x / y))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.95e-16)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.95e-16)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1.95e-16], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999989e-16

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 39.8%

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

   if 1.94999999999999989e-16 < x

   1. Initial program 74.9%

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

   2. Taylor expanded in x around inf 58.7%

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

      1. unpow2 58.7%

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

      2. associate-/l* 61.9%

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

   4. Simplified 61.9%

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

      1. associate-/r/ 61.9%

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

   6. Applied egg-rr 61.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 9: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 1.85e-16) (* 0.5 y) (* 0.5 (/ x (/ y x)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-16) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Fortran

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 (x <= 1.85d-16) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.85e-16) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 1.85e-16:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 1.85e-16)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.85e-16)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 1.85e-16], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-16

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 39.8%

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

   if 1.85e-16 < x

   1. Initial program 74.9%

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

   2. Taylor expanded in x around inf 58.7%

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

      1. unpow2 58.7%

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

      2. associate-/l* 61.9%

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

   4. Simplified 61.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 10: 33.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

2. Taylor expanded in y around inf 30.7%

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

3. Final simplification 30.7%

   \[\leadsto 0.5 \cdot y \]

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 2023279 
(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)))