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

Percentage Accurate: 68.5% → 99.9%

Time: 6.9s

Alternatives: 8

Speedup: 1.2×

• 40.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% 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

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

FPCore

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

C

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

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 = (((x + z) * ((z - x) / y)) - y) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. sub-neg 70.8%

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

   2. +-commutative 70.8%

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

   3. neg-sub0 70.8%

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

   4. associate-+l- 70.8%

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

   5. sub0-neg 70.8%

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

   6. neg-mul-1 70.8%

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

   7. *-commutative 70.8%

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

   8. times-frac 70.8%

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

   9. associate--r+ 70.8%

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

   10. div-sub 70.8%

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

   11. difference-of-squares 76.0%

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

   12. +-commutative 76.0%

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

   13. associate-*r/ 79.0%

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

   14. associate-/l* 99.6%

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

   15. *-inverses 99.6%

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

   16. /-rgt-identity 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{+175}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ y (* x (/ x y))))))
   (if (<= (* z z) 5e+110)
     t_0
     (if (<= (* z z) 1e+175)
       (* -0.5 (* z (/ z y)))
       (if (<= (* z z) 1e+234) t_0 (/ (* z -0.5) (/ y z)))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = 0.5 * (y + (x * (x / y)));
	double tmp;
	if ((z * z) <= 5e+110) {
		tmp = t_0;
	} else if ((z * z) <= 1e+175) {
		tmp = -0.5 * (z * (z / y));
	} else if ((z * z) <= 1e+234) {
		tmp = t_0;
	} else {
		tmp = (z * -0.5) / (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (y + (x * (x / y)))
    if ((z * z) <= 5d+110) then
        tmp = t_0
    else if ((z * z) <= 1d+175) then
        tmp = (-0.5d0) * (z * (z / y))
    else if ((z * z) <= 1d+234) then
        tmp = t_0
    else
        tmp = (z * (-0.5d0)) / (y / z)
    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 * (y + (x * (x / y)));
	double tmp;
	if ((z * z) <= 5e+110) {
		tmp = t_0;
	} else if ((z * z) <= 1e+175) {
		tmp = -0.5 * (z * (z / y));
	} else if ((z * z) <= 1e+234) {
		tmp = t_0;
	} else {
		tmp = (z * -0.5) / (y / z);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = 0.5 * (y + (x * (x / y)))
	tmp = 0
	if (z * z) <= 5e+110:
		tmp = t_0
	elif (z * z) <= 1e+175:
		tmp = -0.5 * (z * (z / y))
	elif (z * z) <= 1e+234:
		tmp = t_0
	else:
		tmp = (z * -0.5) / (y / z)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))))
	tmp = 0.0
	if (Float64(z * z) <= 5e+110)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e+175)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	elseif (Float64(z * z) <= 1e+234)
		tmp = t_0;
	else
		tmp = Float64(Float64(z * -0.5) / Float64(y / z));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (y + (x * (x / y)));
	tmp = 0.0;
	if ((z * z) <= 5e+110)
		tmp = t_0;
	elseif ((z * z) <= 1e+175)
		tmp = -0.5 * (z * (z / y));
	elseif ((z * z) <= 1e+234)
		tmp = t_0;
	else
		tmp = (z * -0.5) / (y / z);
	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[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e+110], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e+175], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+234], t$95$0, N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999978e110 or 9.9999999999999994e174 < (*.f64 z z) < 1.00000000000000002e234

   1. Initial program 74.4%

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

      1. sub-neg 74.4%

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

      2. +-commutative 74.4%

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

      3. neg-sub0 74.4%

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

      4. associate-+l- 74.4%

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

      5. sub0-neg 74.4%

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

      6. neg-mul-1 74.4%

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

      7. *-commutative 74.4%

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

      8. times-frac 74.4%

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

      9. associate--r+ 74.4%

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

      10. div-sub 74.4%

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

      11. difference-of-squares 74.4%

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

      12. +-commutative 74.4%

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

      13. associate-*l/ 78.0%

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

      14. *-commutative 78.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 80.5%

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

      1. sub-neg 80.5%

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

      2. +-commutative 80.5%

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

      3. neg-mul-1 80.5%

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

      4. distribute-lft-out 80.5%

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

      5. associate-*r* 80.5%

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

      6. metadata-eval 80.5%

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

      7. unpow2 80.5%

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

      8. associate-*r/ 87.0%

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

   8. Simplified 87.0%

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

   if 4.99999999999999978e110 < (*.f64 z z) < 9.9999999999999994e174

   1. Initial program 85.6%

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

      1. sub-neg 85.6%

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

      2. +-commutative 85.6%

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

      3. neg-sub0 85.6%

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

      4. associate-+l- 85.6%

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

      5. sub0-neg 85.6%

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

      6. neg-mul-1 85.6%

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

      7. *-commutative 85.6%

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

      8. times-frac 85.6%

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

      9. associate--r+ 85.6%

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

      10. div-sub 85.6%

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

      11. difference-of-squares 85.6%

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

      12. +-commutative 85.6%

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

      13. associate-*r/ 85.6%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 63.5%

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

      1. unpow2 63.5%

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

      2. associate-*l/ 63.5%

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

   6. Simplified 63.5%

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

   if 1.00000000000000002e234 < (*.f64 z z)

   1. Initial program 60.1%

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

      1. sub-neg 60.1%

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

      2. +-commutative 60.1%

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

      3. neg-sub0 60.1%

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

      4. associate-+l- 60.1%

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

      5. sub0-neg 60.1%

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

      6. neg-mul-1 60.1%

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

      7. *-commutative 60.1%

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

      8. times-frac 60.1%

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

      9. associate--r+ 60.1%

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

      10. div-sub 60.1%

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

      11. difference-of-squares 76.9%

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

      12. +-commutative 76.9%

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

      13. associate-*r/ 79.5%

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

      14. associate-/l* 98.9%

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

      15. *-inverses 98.9%

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

      16. /-rgt-identity 98.9%

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

      17. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around inf 75.3%

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

      1. unpow2 75.3%

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

   6. Simplified 75.3%

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

      1. associate-*l/ 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-*l/ 75.3%

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

      4. associate-/l* 78.8%

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

      5. associate-*l/ 79.8%

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

   8. Applied egg-rr 79.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+175}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+234}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \end{array} \]

Alternative 3: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-274)
   (* 0.5 (* x (/ x y)))
   (if (<= (* z z) 1e+25) (* y 0.5) (* -0.5 (* z (/ z y))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-274) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 1e+25) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / 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 ((z * z) <= 5d-274) then
        tmp = 0.5d0 * (x * (x / y))
    else if ((z * z) <= 1d+25) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z * (z / 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 ((z * z) <= 5e-274) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 1e+25) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / y));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-274:
		tmp = 0.5 * (x * (x / y))
	elif (z * z) <= 1e+25:
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z * (z / y))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-274)
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	elseif (Float64(z * z) <= 1e+25)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-274)
		tmp = 0.5 * (x * (x / y));
	elseif ((z * z) <= 1e+25)
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z * (z / y));
	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-274], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+25], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5e-274

   1. Initial program 75.5%

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

      1. sub-neg 75.5%

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

      2. +-commutative 75.5%

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

      3. neg-sub0 75.5%

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

      4. associate-+l- 75.5%

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

      5. sub0-neg 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. times-frac 75.5%

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

      9. associate--r+ 75.5%

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

      10. div-sub 75.5%

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

      11. difference-of-squares 75.5%

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

      12. +-commutative 75.5%

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

      13. associate-*l/ 82.4%

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

      14. *-commutative 82.4%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. unpow2 47.8%

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

      3. associate-*r/ 54.6%

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

   8. Simplified 54.6%

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

   if 5e-274 < (*.f64 z z) < 1.00000000000000009e25

   1. Initial program 70.7%

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

      1. sub-neg 70.7%

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

      2. +-commutative 70.7%

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

      3. neg-sub0 70.7%

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

      4. associate-+l- 70.7%

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

      5. sub0-neg 70.7%

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

      6. neg-mul-1 70.7%

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

      7. *-commutative 70.7%

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

      8. times-frac 70.7%

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

      9. associate--r+ 70.7%

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

      10. div-sub 70.7%

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

      11. difference-of-squares 70.7%

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

      12. +-commutative 70.7%

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

      13. associate-*r/ 70.8%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 62.9%

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

      1. *-commutative 62.9%

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

   6. Simplified 62.9%

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

   if 1.00000000000000009e25 < (*.f64 z z)

   1. Initial program 67.9%

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

      1. sub-neg 67.9%

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

      2. +-commutative 67.9%

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

      3. neg-sub0 67.9%

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

      4. associate-+l- 67.9%

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

      5. sub0-neg 67.9%

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

      6. neg-mul-1 67.9%

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

      7. *-commutative 67.9%

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

      8. times-frac 67.9%

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

      9. associate--r+ 67.9%

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

      10. div-sub 67.9%

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

      11. difference-of-squares 78.1%

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

      12. +-commutative 78.1%

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

      13. associate-*r/ 79.7%

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

      14. associate-/l* 99.3%

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

      15. *-inverses 99.3%

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

      16. /-rgt-identity 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around inf 62.4%

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

      1. unpow2 62.4%

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

      2. associate-*l/ 64.5%

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

   6. Simplified 64.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 4: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-274)
   (* 0.5 (* x (/ x y)))
   (if (<= (* z z) 1e+25) (* y 0.5) (/ (* z -0.5) (/ y z)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-274) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 1e+25) {
		tmp = y * 0.5;
	} else {
		tmp = (z * -0.5) / (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) <= 5d-274) then
        tmp = 0.5d0 * (x * (x / y))
    else if ((z * z) <= 1d+25) then
        tmp = y * 0.5d0
    else
        tmp = (z * (-0.5d0)) / (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) <= 5e-274) {
		tmp = 0.5 * (x * (x / y));
	} else if ((z * z) <= 1e+25) {
		tmp = y * 0.5;
	} else {
		tmp = (z * -0.5) / (y / z);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-274:
		tmp = 0.5 * (x * (x / y))
	elif (z * z) <= 1e+25:
		tmp = y * 0.5
	else:
		tmp = (z * -0.5) / (y / z)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-274)
		tmp = Float64(0.5 * Float64(x * Float64(x / y)));
	elseif (Float64(z * z) <= 1e+25)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(z * -0.5) / Float64(y / z));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-274)
		tmp = 0.5 * (x * (x / y));
	elseif ((z * z) <= 1e+25)
		tmp = y * 0.5;
	else
		tmp = (z * -0.5) / (y / 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], 5e-274], N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+25], N[(y * 0.5), $MachinePrecision], N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5e-274

   1. Initial program 75.5%

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

      1. sub-neg 75.5%

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

      2. +-commutative 75.5%

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

      3. neg-sub0 75.5%

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

      4. associate-+l- 75.5%

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

      5. sub0-neg 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. times-frac 75.5%

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

      9. associate--r+ 75.5%

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

      10. div-sub 75.5%

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

      11. difference-of-squares 75.5%

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

      12. +-commutative 75.5%

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

      13. associate-*l/ 82.4%

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

      14. *-commutative 82.4%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. unpow2 47.8%

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

      3. associate-*r/ 54.6%

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

   8. Simplified 54.6%

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

   if 5e-274 < (*.f64 z z) < 1.00000000000000009e25

   1. Initial program 70.7%

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

      1. sub-neg 70.7%

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

      2. +-commutative 70.7%

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

      3. neg-sub0 70.7%

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

      4. associate-+l- 70.7%

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

      5. sub0-neg 70.7%

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

      6. neg-mul-1 70.7%

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

      7. *-commutative 70.7%

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

      8. times-frac 70.7%

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

      9. associate--r+ 70.7%

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

      10. div-sub 70.7%

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

      11. difference-of-squares 70.7%

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

      12. +-commutative 70.7%

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

      13. associate-*r/ 70.8%

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

      14. associate-/l* 100.0%

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

      15. *-inverses 100.0%

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

      16. /-rgt-identity 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 62.9%

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

      1. *-commutative 62.9%

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

   6. Simplified 62.9%

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

   if 1.00000000000000009e25 < (*.f64 z z)

   1. Initial program 67.9%

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

      1. sub-neg 67.9%

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

      2. +-commutative 67.9%

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

      3. neg-sub0 67.9%

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

      4. associate-+l- 67.9%

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

      5. sub0-neg 67.9%

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

      6. neg-mul-1 67.9%

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

      7. *-commutative 67.9%

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

      8. times-frac 67.9%

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

      9. associate--r+ 67.9%

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

      10. div-sub 67.9%

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

      11. difference-of-squares 78.1%

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

      12. +-commutative 78.1%

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

      13. associate-*r/ 79.7%

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

      14. associate-/l* 99.3%

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

      15. *-inverses 99.3%

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

      16. /-rgt-identity 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around inf 62.4%

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

      1. unpow2 62.4%

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

   6. Simplified 62.4%

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

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

      3. associate-*l/ 62.4%

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

      4. associate-/l* 64.5%

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

      5. associate-*l/ 65.1%

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

   8. Applied egg-rr 65.1%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+25}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \end{array} \]

Alternative 5: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e+111)
   (* -0.5 (- (* z (/ z y)) y))
   (* -0.5 (- (* (- z x) (/ x y)) y))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+111) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - 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 * x) <= 5d+111) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (-0.5d0) * (((z - x) * (x / y)) - 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 * x) <= 5e+111) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e+111:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+111)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e+111)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+111], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999997e111

   1. Initial program 74.3%

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

      1. sub-neg 74.3%

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

      2. +-commutative 74.3%

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

      3. neg-sub0 74.3%

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

      4. associate-+l- 74.3%

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

      5. sub0-neg 74.3%

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

      6. neg-mul-1 74.3%

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

      7. *-commutative 74.3%

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

      8. times-frac 74.3%

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

      9. associate--r+ 74.3%

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

      10. div-sub 74.3%

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

      11. difference-of-squares 74.3%

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

      12. +-commutative 74.3%

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

      13. associate-*r/ 76.9%

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

      14. associate-/l* 99.4%

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

      15. *-inverses 99.4%

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

      16. /-rgt-identity 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 88.7%

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

      1. unpow2 88.7%

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

      2. associate-*l/ 93.7%

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

   6. Simplified 93.7%

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

   if 4.9999999999999997e111 < (*.f64 x x)

   1. Initial program 65.7%

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

      1. sub-neg 65.7%

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

      2. +-commutative 65.7%

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

      3. neg-sub0 65.7%

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

      4. associate-+l- 65.7%

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

      5. sub0-neg 65.7%

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

      6. neg-mul-1 65.7%

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

      7. *-commutative 65.7%

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

      8. times-frac 65.7%

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

      9. associate--r+ 65.7%

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

      10. div-sub 65.7%

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

      11. difference-of-squares 78.4%

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

      12. +-commutative 78.4%

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

      13. associate-*l/ 82.0%

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

      14. *-commutative 82.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 87.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+184}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+184)
   (* -0.5 (- (* z (/ z y)) y))
   (* 0.5 (+ y (* x (/ x y))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+184) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = 0.5 * (y + (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 * x) <= 1d+184) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = 0.5d0 * (y + (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 * x) <= 1e+184) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = 0.5 * (y + (x * (x / y)));
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+184:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = 0.5 * (y + (x * (x / y)))
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+184)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(0.5 * Float64(y + 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 * x) <= 1e+184)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = 0.5 * (y + (x * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+184], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000002e184

   1. Initial program 75.3%

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

      1. sub-neg 75.3%

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

      2. +-commutative 75.3%

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

      3. neg-sub0 75.3%

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

      4. associate-+l- 75.3%

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

      5. sub0-neg 75.3%

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

      6. neg-mul-1 75.3%

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

      7. *-commutative 75.3%

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

      8. times-frac 75.3%

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

      9. associate--r+ 75.3%

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

      10. div-sub 75.3%

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

      11. difference-of-squares 75.3%

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

      12. +-commutative 75.3%

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

      13. associate-*r/ 77.7%

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

      14. associate-/l* 99.4%

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

      15. *-inverses 99.4%

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

      16. /-rgt-identity 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 87.9%

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

      1. unpow2 87.9%

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

      2. associate-*l/ 92.5%

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

   6. Simplified 92.5%

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

   if 1.00000000000000002e184 < (*.f64 x x)

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. +-commutative 62.5%

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

      3. neg-sub0 62.5%

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

      4. associate-+l- 62.5%

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

      5. sub0-neg 62.5%

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

      6. neg-mul-1 62.5%

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

      7. *-commutative 62.5%

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

      8. times-frac 62.5%

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

      9. associate--r+ 62.5%

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

      10. div-sub 62.5%

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

      11. difference-of-squares 77.3%

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

      12. +-commutative 77.3%

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

      13. associate-*l/ 81.4%

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

      14. *-commutative 81.4%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 70.4%

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

      1. sub-neg 70.4%

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

      2. +-commutative 70.4%

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

      3. neg-mul-1 70.4%

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

      4. distribute-lft-out 70.4%

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

      5. associate-*r* 70.4%

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

      6. metadata-eval 70.4%

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

      7. unpow2 70.4%

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

      8. associate-*r/ 86.6%

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

   8. Simplified 86.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+184}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 7: 54.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+79}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y -8.2e+79)
   (* y 0.5)
   (if (<= y 5.8e+44) (* -0.5 (* z (/ z y))) (* y 0.5))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+79) {
		tmp = y * 0.5;
	} else if (y <= 5.8e+44) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = 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 (y <= (-8.2d+79)) then
        tmp = y * 0.5d0
    else if (y <= 5.8d+44) then
        tmp = (-0.5d0) * (z * (z / y))
    else
        tmp = 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 (y <= -8.2e+79) {
		tmp = y * 0.5;
	} else if (y <= 5.8e+44) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if y <= -8.2e+79:
		tmp = y * 0.5
	elif y <= 5.8e+44:
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = y * 0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (y <= -8.2e+79)
		tmp = Float64(y * 0.5);
	elseif (y <= 5.8e+44)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.2e+79)
		tmp = y * 0.5;
	elseif (y <= 5.8e+44)
		tmp = -0.5 * (z * (z / y));
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, -8.2e+79], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 5.8e+44], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2e79 or 5.8000000000000004e44 < y

   1. Initial program 42.3%

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

      1. sub-neg 42.3%

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

      2. +-commutative 42.3%

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

      3. neg-sub0 42.3%

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

      4. associate-+l- 42.3%

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

      5. sub0-neg 42.3%

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

      6. neg-mul-1 42.3%

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

      7. *-commutative 42.3%

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

      8. times-frac 42.3%

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

      9. associate--r+ 42.3%

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

      10. div-sub 42.3%

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

      11. difference-of-squares 44.4%

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

      12. +-commutative 44.4%

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

      13. associate-*r/ 49.1%

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

      14. associate-/l* 99.2%

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

      15. *-inverses 99.2%

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

      16. /-rgt-identity 99.2%

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

      17. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 69.1%

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

      1. *-commutative 69.1%

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

   6. Simplified 69.1%

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

   if -8.2e79 < y < 5.8000000000000004e44

   1. Initial program 90.7%

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

      1. sub-neg 90.7%

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

      2. +-commutative 90.7%

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

      3. neg-sub0 90.7%

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

      4. associate-+l- 90.7%

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

      5. sub0-neg 90.7%

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

      6. neg-mul-1 90.7%

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

      7. *-commutative 90.7%

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

      8. times-frac 90.7%

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

      9. associate--r+ 90.7%

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

      10. div-sub 90.7%

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

      11. difference-of-squares 98.0%

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

      12. +-commutative 98.0%

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

      13. associate-*r/ 99.8%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 54.0%

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

      1. unpow2 54.0%

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

      2. associate-*l/ 54.5%

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

   6. Simplified 54.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+79}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+44}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8: 34.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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 = y * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

   1. sub-neg 70.8%

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

   2. +-commutative 70.8%

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

   3. neg-sub0 70.8%

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

   4. associate-+l- 70.8%

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

   5. sub0-neg 70.8%

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

   6. neg-mul-1 70.8%

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

   7. *-commutative 70.8%

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

   8. times-frac 70.8%

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

   9. associate--r+ 70.8%

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

   10. div-sub 70.8%

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

   11. difference-of-squares 76.0%

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

   12. +-commutative 76.0%

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

   13. associate-*r/ 79.0%

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

   14. associate-/l* 99.6%

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

   15. *-inverses 99.6%

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

   16. /-rgt-identity 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around inf 33.6%

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

   1. *-commutative 33.6%

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

6. Simplified 33.6%

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

7. Final simplification 33.6%

   \[\leadsto y \cdot 0.5 \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(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)))