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

Percentage Accurate: 69.6% → 99.9%

Time: 8.4s

Alternatives: 11

Speedup: 1.2×

• 42.1% 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: 69.6% 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 70.3%

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

   1. sub-neg 70.3%

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

   2. +-commutative 70.3%

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

   3. neg-sub0 70.3%

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

   4. associate-+l- 70.3%

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

   5. sub0-neg 70.3%

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

   6. neg-mul-1 70.3%

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

   7. *-commutative 70.3%

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

   8. times-frac 70.3%

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

   9. associate--r+ 70.3%

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

   10. div-sub 70.3%

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

   11. difference-of-squares 74.6%

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

   12. +-commutative 74.6%

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

   13. associate-*r/ 77.9%

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

   14. associate-/l* 99.9%

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

   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. Final simplification 99.9%

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

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+101} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+161}\right):\\ \;\;\;\;-0.5 \cdot \left(\frac{-x}{\frac{y}{x}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-68)
   (* -0.5 (- (* z (/ z y)) y))
   (if (or (<= (* x x) 2e+101) (not (<= (* x x) 2e+161)))
     (* -0.5 (- (/ (- x) (/ y x)) y))
     (* -0.5 (- (/ z (/ y z)) y)))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-68) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else if (((x * x) <= 2e+101) || !((x * x) <= 2e+161)) {
		tmp = -0.5 * ((-x / (y / x)) - y);
	} else {
		tmp = -0.5 * ((z / (y / 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 ((x * x) <= 1d-68) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else if (((x * x) <= 2d+101) .or. (.not. ((x * x) <= 2d+161))) then
        tmp = (-0.5d0) * ((-x / (y / x)) - y)
    else
        tmp = (-0.5d0) * ((z / (y / 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 ((x * x) <= 1e-68) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else if (((x * x) <= 2e+101) || !((x * x) <= 2e+161)) {
		tmp = -0.5 * ((-x / (y / x)) - y);
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-68:
		tmp = -0.5 * ((z * (z / y)) - y)
	elif ((x * x) <= 2e+101) or not ((x * x) <= 2e+161):
		tmp = -0.5 * ((-x / (y / x)) - y)
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-68)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	elseif ((Float64(x * x) <= 2e+101) || !(Float64(x * x) <= 2e+161))
		tmp = Float64(-0.5 * Float64(Float64(Float64(-x) / Float64(y / x)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-68)
		tmp = -0.5 * ((z * (z / y)) - y);
	elseif (((x * x) <= 2e+101) || ~(((x * x) <= 2e+161)))
		tmp = -0.5 * ((-x / (y / x)) - y);
	else
		tmp = -0.5 * ((z / (y / z)) - 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-68], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * x), $MachinePrecision], 2e+101], N[Not[LessEqual[N[(x * x), $MachinePrecision], 2e+161]], $MachinePrecision]], N[(-0.5 * N[(N[((-x) / N[(y / x), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.00000000000000007e-68

   1. Initial program 70.6%

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

      1. sub-neg 70.6%

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

      2. +-commutative 70.6%

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

      3. neg-sub0 70.6%

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

      4. associate-+l- 70.6%

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

      5. sub0-neg 70.6%

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

      6. neg-mul-1 70.6%

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

      7. *-commutative 70.6%

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

      8. times-frac 70.6%

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

      9. associate--r+ 70.6%

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

      10. div-sub 70.6%

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

      11. difference-of-squares 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-*l/ 75.6%

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

      14. *-commutative 75.6%

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

      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 99.8%

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

   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 z around inf 82.7%

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

      1. unpow2 82.7%

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

      2. associate-*r/ 89.4%

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

   8. Simplified 89.4%

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

   if 1.00000000000000007e-68 < (*.f64 x x) < 2e101 or 2.0000000000000001e161 < (*.f64 x x)

   1. Initial program 70.2%

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

      1. sub-neg 70.2%

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

      2. +-commutative 70.2%

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

      3. neg-sub0 70.2%

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

      4. associate-+l- 70.2%

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

      5. sub0-neg 70.2%

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

      6. neg-mul-1 70.2%

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

      7. *-commutative 70.2%

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

      8. times-frac 70.2%

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

      9. associate--r+ 70.2%

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

      10. div-sub 70.2%

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

      11. difference-of-squares 80.1%

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

      12. +-commutative 80.1%

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

      13. associate-*l/ 81.8%

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

      14. *-commutative 81.8%

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

      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 75.6%

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

      1. mul-1-neg 75.6%

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

      2. unpow2 75.6%

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

      3. associate-/l* 83.9%

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

      4. distribute-neg-frac 83.9%

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

   8. Simplified 83.9%

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

   if 2e101 < (*.f64 x x) < 2.0000000000000001e161

   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 67.9%

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

      12. +-commutative 67.9%

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

      13. associate-*l/ 67.9%

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

      14. *-commutative 67.9%

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

      15. associate-/l* 99.6%

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

      16. *-inverses 99.6%

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

      17. /-rgt-identity 99.6%

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

      18. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 50.0%

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

      3. times-frac 49.8%

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

      4. +-commutative 49.8%

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

      5. +-commutative 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. associate-*l/ 49.8%

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

      2. *-lft-identity 49.8%

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

      3. +-commutative 49.8%

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

      4. +-commutative 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in z around inf 68.3%

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

       1. unpow2 68.3%

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

       2. associate-/l* 83.7%

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

   12. Simplified 83.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+101} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+161}\right):\\ \;\;\;\;-0.5 \cdot \left(\frac{-x}{\frac{y}{x}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(\frac{-x \cdot x}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+161}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{-x}{\frac{y}{x}} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-68)
   (* -0.5 (- (* z (/ z y)) y))
   (if (<= (* x x) 2e+101)
     (* -0.5 (- (/ (- (* x x)) y) y))
     (if (<= (* x x) 2e+161)
       (* -0.5 (- (/ z (/ y z)) y))
       (* -0.5 (- (/ (- x) (/ y x)) y))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-68) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else if ((x * x) <= 2e+101) {
		tmp = -0.5 * ((-(x * x) / y) - y);
	} else if ((x * x) <= 2e+161) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * ((-x / (y / 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-68) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else if ((x * x) <= 2d+101) then
        tmp = (-0.5d0) * ((-(x * x) / y) - y)
    else if ((x * x) <= 2d+161) then
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    else
        tmp = (-0.5d0) * ((-x / (y / 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-68) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else if ((x * x) <= 2e+101) {
		tmp = -0.5 * ((-(x * x) / y) - y);
	} else if ((x * x) <= 2e+161) {
		tmp = -0.5 * ((z / (y / z)) - y);
	} else {
		tmp = -0.5 * ((-x / (y / x)) - y);
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-68:
		tmp = -0.5 * ((z * (z / y)) - y)
	elif (x * x) <= 2e+101:
		tmp = -0.5 * ((-(x * x) / y) - y)
	elif (x * x) <= 2e+161:
		tmp = -0.5 * ((z / (y / z)) - y)
	else:
		tmp = -0.5 * ((-x / (y / x)) - y)
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-68)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	elseif (Float64(x * x) <= 2e+101)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-Float64(x * x)) / y) - y));
	elseif (Float64(x * x) <= 2e+161)
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-x) / Float64(y / x)) - y));
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-68)
		tmp = -0.5 * ((z * (z / y)) - y);
	elseif ((x * x) <= 2e+101)
		tmp = -0.5 * ((-(x * x) / y) - y);
	elseif ((x * x) <= 2e+161)
		tmp = -0.5 * ((z / (y / z)) - y);
	else
		tmp = -0.5 * ((-x / (y / 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-68], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+101], N[(-0.5 * N[(N[((-N[(x * x), $MachinePrecision]) / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+161], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[((-x) / N[(y / x), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000000000000007e-68

   1. Initial program 70.6%

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

      1. sub-neg 70.6%

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

      2. +-commutative 70.6%

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

      3. neg-sub0 70.6%

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

      4. associate-+l- 70.6%

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

      5. sub0-neg 70.6%

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

      6. neg-mul-1 70.6%

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

      7. *-commutative 70.6%

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

      8. times-frac 70.6%

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

      9. associate--r+ 70.6%

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

      10. div-sub 70.6%

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

      11. difference-of-squares 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-*l/ 75.6%

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

      14. *-commutative 75.6%

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

      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 99.8%

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

   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 z around inf 82.7%

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

      1. unpow2 82.7%

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

      2. associate-*r/ 89.4%

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

   8. Simplified 89.4%

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

   if 1.00000000000000007e-68 < (*.f64 x x) < 2e101

   1. Initial program 86.3%

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

      1. sub-neg 86.3%

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

      2. +-commutative 86.3%

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

      3. neg-sub0 86.3%

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

      4. associate-+l- 86.3%

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

      5. sub0-neg 86.3%

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

      6. neg-mul-1 86.3%

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

      7. *-commutative 86.3%

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

      8. times-frac 86.3%

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

      9. associate--r+ 86.3%

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

      10. div-sub 86.3%

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

      11. difference-of-squares 86.3%

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

      12. +-commutative 86.3%

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

      13. associate-*r/ 86.3%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 84.8%

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

      1. unpow2 84.8%

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

      2. associate-*r/ 84.8%

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

      3. neg-mul-1 84.8%

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

      4. distribute-rgt-neg-in 84.8%

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

   6. Simplified 84.8%

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

   if 2e101 < (*.f64 x x) < 2.0000000000000001e161

   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 67.9%

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

      12. +-commutative 67.9%

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

      13. associate-*l/ 67.9%

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

      14. *-commutative 67.9%

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

      15. associate-/l* 99.6%

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

      16. *-inverses 99.6%

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

      17. /-rgt-identity 99.6%

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

      18. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 50.0%

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

      3. times-frac 49.8%

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

      4. +-commutative 49.8%

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

      5. +-commutative 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. associate-*l/ 49.8%

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

      2. *-lft-identity 49.8%

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

      3. +-commutative 49.8%

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

      4. +-commutative 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in z around inf 68.3%

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

       1. unpow2 68.3%

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

       2. associate-/l* 83.7%

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

   12. Simplified 83.7%

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

   if 2.0000000000000001e161 < (*.f64 x x)

   1. Initial program 63.4%

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

      1. sub-neg 63.4%

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

      2. +-commutative 63.4%

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

      3. neg-sub0 63.4%

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

      4. associate-+l- 63.4%

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

      5. sub0-neg 63.4%

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

      6. neg-mul-1 63.4%

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

      7. *-commutative 63.4%

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

      8. times-frac 63.4%

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

      9. associate--r+ 63.4%

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

      10. div-sub 63.4%

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

      11. difference-of-squares 77.5%

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

      12. +-commutative 77.5%

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

      13. associate-*l/ 79.9%

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

      14. *-commutative 79.9%

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

      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 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around 0 71.8%

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

      1. mul-1-neg 71.8%

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

      2. unpow2 71.8%

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

      3. associate-/l* 83.5%

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

      4. distribute-neg-frac 83.5%

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

   8. Simplified 83.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(\frac{-x \cdot x}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+161}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{-x}{\frac{y}{x}} - y\right)\\ \end{array} \]

Alternative 4: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ t_1 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))) (t_1 (* x (* x (/ 0.5 y)))))
   (if (<= y -2.15e+49)
     (* y 0.5)
     (if (<= y -1.45e-12)
       t_1
       (if (<= y -9.8e-21)
         (* y 0.5)
         (if (<= y -4.1e-81)
           t_0
           (if (<= y -3.3e-258)
             t_1
             (if (<= y 1.1e-43) t_0 (if (<= y 1.05e+121) t_1 (* y 0.5))))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= -2.15e+49) {
		tmp = y * 0.5;
	} else if (y <= -1.45e-12) {
		tmp = t_1;
	} else if (y <= -9.8e-21) {
		tmp = y * 0.5;
	} else if (y <= -4.1e-81) {
		tmp = t_0;
	} else if (y <= -3.3e-258) {
		tmp = t_1;
	} else if (y <= 1.1e-43) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    t_1 = x * (x * (0.5d0 / y))
    if (y <= (-2.15d+49)) then
        tmp = y * 0.5d0
    else if (y <= (-1.45d-12)) then
        tmp = t_1
    else if (y <= (-9.8d-21)) then
        tmp = y * 0.5d0
    else if (y <= (-4.1d-81)) then
        tmp = t_0
    else if (y <= (-3.3d-258)) then
        tmp = t_1
    else if (y <= 1.1d-43) then
        tmp = t_0
    else if (y <= 1.05d+121) then
        tmp = t_1
    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 t_0 = -0.5 * (z * (z / y));
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= -2.15e+49) {
		tmp = y * 0.5;
	} else if (y <= -1.45e-12) {
		tmp = t_1;
	} else if (y <= -9.8e-21) {
		tmp = y * 0.5;
	} else if (y <= -4.1e-81) {
		tmp = t_0;
	} else if (y <= -3.3e-258) {
		tmp = t_1;
	} else if (y <= 1.1e-43) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = t_1;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	t_1 = x * (x * (0.5 / y))
	tmp = 0
	if y <= -2.15e+49:
		tmp = y * 0.5
	elif y <= -1.45e-12:
		tmp = t_1
	elif y <= -9.8e-21:
		tmp = y * 0.5
	elif y <= -4.1e-81:
		tmp = t_0
	elif y <= -3.3e-258:
		tmp = t_1
	elif y <= 1.1e-43:
		tmp = t_0
	elif y <= 1.05e+121:
		tmp = t_1
	else:
		tmp = y * 0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	t_1 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (y <= -2.15e+49)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.45e-12)
		tmp = t_1;
	elseif (y <= -9.8e-21)
		tmp = Float64(y * 0.5);
	elseif (y <= -4.1e-81)
		tmp = t_0;
	elseif (y <= -3.3e-258)
		tmp = t_1;
	elseif (y <= 1.1e-43)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = t_1;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	t_1 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (y <= -2.15e+49)
		tmp = y * 0.5;
	elseif (y <= -1.45e-12)
		tmp = t_1;
	elseif (y <= -9.8e-21)
		tmp = y * 0.5;
	elseif (y <= -4.1e-81)
		tmp = t_0;
	elseif (y <= -3.3e-258)
		tmp = t_1;
	elseif (y <= 1.1e-43)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = t_1;
	else
		tmp = y * 0.5;
	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[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.15e+49], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.45e-12], t$95$1, If[LessEqual[y, -9.8e-21], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -4.1e-81], t$95$0, If[LessEqual[y, -3.3e-258], t$95$1, If[LessEqual[y, 1.1e-43], t$95$0, If[LessEqual[y, 1.05e+121], t$95$1, N[(y * 0.5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.15e49 or -1.4500000000000001e-12 < y < -9.8000000000000003e-21 or 1.0500000000000001e121 < y

   1. Initial program 41.8%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if -2.15e49 < y < -1.4500000000000001e-12 or -4.09999999999999984e-81 < y < -3.3e-258 or 1.09999999999999999e-43 < y < 1.0500000000000001e121

   1. Initial program 91.9%

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

   2. Taylor expanded in x around inf 57.4%

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

      1. unpow2 57.4%

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

   4. Simplified 57.4%

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

   5. Taylor expanded in x around 0 57.4%

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

      1. unpow2 57.4%

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

      2. associate-*r/ 57.4%

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

      3. associate-*l/ 57.4%

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

      4. *-commutative 57.4%

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

      5. associate-*r* 59.5%

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

   7. Simplified 59.5%

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

   if -9.8000000000000003e-21 < y < -4.09999999999999984e-81 or -3.3e-258 < y < 1.09999999999999999e-43

   1. Initial program 84.0%

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

   2. Taylor expanded in y around 0 81.8%

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

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in x around 0 60.9%

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

      1. unpow2 60.9%

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

      2. associate-*l/ 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+49}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ t_1 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))) (t_1 (* x (* x (/ 0.5 y)))))
   (if (<= y -4.5e+48)
     (* y 0.5)
     (if (<= y -1.8e-11)
       t_1
       (if (<= y -1.45e-21)
         (* y 0.5)
         (if (<= y -3.7e-81)
           t_0
           (if (<= y -9e-258)
             (* x (/ x (* y 2.0)))
             (if (<= y 1.3e-42) t_0 (if (<= y 1.05e+121) t_1 (* y 0.5))))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= -4.5e+48) {
		tmp = y * 0.5;
	} else if (y <= -1.8e-11) {
		tmp = t_1;
	} else if (y <= -1.45e-21) {
		tmp = y * 0.5;
	} else if (y <= -3.7e-81) {
		tmp = t_0;
	} else if (y <= -9e-258) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 1.3e-42) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    t_1 = x * (x * (0.5d0 / y))
    if (y <= (-4.5d+48)) then
        tmp = y * 0.5d0
    else if (y <= (-1.8d-11)) then
        tmp = t_1
    else if (y <= (-1.45d-21)) then
        tmp = y * 0.5d0
    else if (y <= (-3.7d-81)) then
        tmp = t_0
    else if (y <= (-9d-258)) then
        tmp = x * (x / (y * 2.0d0))
    else if (y <= 1.3d-42) then
        tmp = t_0
    else if (y <= 1.05d+121) then
        tmp = t_1
    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 t_0 = -0.5 * (z * (z / y));
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= -4.5e+48) {
		tmp = y * 0.5;
	} else if (y <= -1.8e-11) {
		tmp = t_1;
	} else if (y <= -1.45e-21) {
		tmp = y * 0.5;
	} else if (y <= -3.7e-81) {
		tmp = t_0;
	} else if (y <= -9e-258) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 1.3e-42) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = t_1;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	t_1 = x * (x * (0.5 / y))
	tmp = 0
	if y <= -4.5e+48:
		tmp = y * 0.5
	elif y <= -1.8e-11:
		tmp = t_1
	elif y <= -1.45e-21:
		tmp = y * 0.5
	elif y <= -3.7e-81:
		tmp = t_0
	elif y <= -9e-258:
		tmp = x * (x / (y * 2.0))
	elif y <= 1.3e-42:
		tmp = t_0
	elif y <= 1.05e+121:
		tmp = t_1
	else:
		tmp = y * 0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	t_1 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (y <= -4.5e+48)
		tmp = Float64(y * 0.5);
	elseif (y <= -1.8e-11)
		tmp = t_1;
	elseif (y <= -1.45e-21)
		tmp = Float64(y * 0.5);
	elseif (y <= -3.7e-81)
		tmp = t_0;
	elseif (y <= -9e-258)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (y <= 1.3e-42)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = t_1;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	t_1 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (y <= -4.5e+48)
		tmp = y * 0.5;
	elseif (y <= -1.8e-11)
		tmp = t_1;
	elseif (y <= -1.45e-21)
		tmp = y * 0.5;
	elseif (y <= -3.7e-81)
		tmp = t_0;
	elseif (y <= -9e-258)
		tmp = x * (x / (y * 2.0));
	elseif (y <= 1.3e-42)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = t_1;
	else
		tmp = y * 0.5;
	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[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+48], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -1.8e-11], t$95$1, If[LessEqual[y, -1.45e-21], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -3.7e-81], t$95$0, If[LessEqual[y, -9e-258], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-42], t$95$0, If[LessEqual[y, 1.05e+121], t$95$1, N[(y * 0.5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.49999999999999995e48 or -1.79999999999999992e-11 < y < -1.45e-21 or 1.0500000000000001e121 < y

   1. Initial program 41.8%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if -4.49999999999999995e48 < y < -1.79999999999999992e-11 or 1.3e-42 < y < 1.0500000000000001e121

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf 54.5%

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

      1. unpow2 54.5%

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

   4. Simplified 54.5%

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

   5. Taylor expanded in x around 0 54.5%

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

      1. unpow2 54.5%

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

      2. associate-*r/ 54.5%

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

      3. associate-*l/ 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*r* 57.1%

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

   7. Simplified 57.1%

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

   if -1.45e-21 < y < -3.69999999999999986e-81 or -9.00000000000000017e-258 < y < 1.3e-42

   1. Initial program 84.0%

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

   2. Taylor expanded in y around 0 81.8%

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

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in x around 0 60.9%

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

      1. unpow2 60.9%

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

      2. associate-*l/ 63.4%

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

   7. Simplified 63.4%

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

   if -3.69999999999999986e-81 < y < -9.00000000000000017e-258

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 59.9%

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

      1. unpow2 59.9%

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

   4. Simplified 59.9%

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

      1. associate-/l* 61.6%

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

      2. associate-/r/ 61.6%

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

   6. Applied egg-rr 61.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 51.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.25 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= y -9.2e+48)
     (* y 0.5)
     (if (<= y -6.2e-8)
       (* x (* x (/ 0.5 y)))
       (if (<= y -1.35e-21)
         (* y 0.5)
         (if (<= y -3.25e-81)
           t_0
           (if (<= y -6.5e-258)
             (* x (/ x (* y 2.0)))
             (if (<= y 9.8e-43)
               t_0
               (if (<= y 1.05e+121) (* (/ x (/ y x)) 0.5) (* y 0.5))))))))))

C

z = abs(z);
double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -9.2e+48) {
		tmp = y * 0.5;
	} else if (y <= -6.2e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= -1.35e-21) {
		tmp = y * 0.5;
	} else if (y <= -3.25e-81) {
		tmp = t_0;
	} else if (y <= -6.5e-258) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 9.8e-43) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = (x / (y / x)) * 0.5;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    if (y <= (-9.2d+48)) then
        tmp = y * 0.5d0
    else if (y <= (-6.2d-8)) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= (-1.35d-21)) then
        tmp = y * 0.5d0
    else if (y <= (-3.25d-81)) then
        tmp = t_0
    else if (y <= (-6.5d-258)) then
        tmp = x * (x / (y * 2.0d0))
    else if (y <= 9.8d-43) then
        tmp = t_0
    else if (y <= 1.05d+121) then
        tmp = (x / (y / x)) * 0.5d0
    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 t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -9.2e+48) {
		tmp = y * 0.5;
	} else if (y <= -6.2e-8) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= -1.35e-21) {
		tmp = y * 0.5;
	} else if (y <= -3.25e-81) {
		tmp = t_0;
	} else if (y <= -6.5e-258) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 9.8e-43) {
		tmp = t_0;
	} else if (y <= 1.05e+121) {
		tmp = (x / (y / x)) * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if y <= -9.2e+48:
		tmp = y * 0.5
	elif y <= -6.2e-8:
		tmp = x * (x * (0.5 / y))
	elif y <= -1.35e-21:
		tmp = y * 0.5
	elif y <= -3.25e-81:
		tmp = t_0
	elif y <= -6.5e-258:
		tmp = x * (x / (y * 2.0))
	elif y <= 9.8e-43:
		tmp = t_0
	elif y <= 1.05e+121:
		tmp = (x / (y / x)) * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (y <= -9.2e+48)
		tmp = Float64(y * 0.5);
	elseif (y <= -6.2e-8)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= -1.35e-21)
		tmp = Float64(y * 0.5);
	elseif (y <= -3.25e-81)
		tmp = t_0;
	elseif (y <= -6.5e-258)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (y <= 9.8e-43)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = Float64(Float64(x / Float64(y / x)) * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (y <= -9.2e+48)
		tmp = y * 0.5;
	elseif (y <= -6.2e-8)
		tmp = x * (x * (0.5 / y));
	elseif (y <= -1.35e-21)
		tmp = y * 0.5;
	elseif (y <= -3.25e-81)
		tmp = t_0;
	elseif (y <= -6.5e-258)
		tmp = x * (x / (y * 2.0));
	elseif (y <= 9.8e-43)
		tmp = t_0;
	elseif (y <= 1.05e+121)
		tmp = (x / (y / x)) * 0.5;
	else
		tmp = y * 0.5;
	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[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.2e+48], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -6.2e-8], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.35e-21], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, -3.25e-81], t$95$0, If[LessEqual[y, -6.5e-258], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e-43], t$95$0, If[LessEqual[y, 1.05e+121], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -9.2000000000000001e48 or -6.2e-8 < y < -1.3500000000000001e-21 or 1.0500000000000001e121 < y

   1. Initial program 41.8%

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

   2. Taylor expanded in y around inf 74.0%

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

      1. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if -9.2000000000000001e48 < y < -6.2e-8

   1. Initial program 88.7%

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

   2. Taylor expanded in x around inf 66.9%

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

      1. unpow2 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in x around 0 66.9%

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

      1. unpow2 66.9%

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

      2. associate-*r/ 66.9%

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

      3. associate-*l/ 66.9%

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

      4. *-commutative 66.9%

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

      5. associate-*r* 67.1%

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

   7. Simplified 67.1%

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

   if -1.3500000000000001e-21 < y < -3.2500000000000001e-81 or -6.5000000000000002e-258 < y < 9.79999999999999976e-43

   1. Initial program 84.0%

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

   2. Taylor expanded in y around 0 81.8%

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

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

   4. Simplified 81.8%

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

   5. Taylor expanded in x around 0 60.9%

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

      1. unpow2 60.9%

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

      2. associate-*l/ 63.4%

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

   7. Simplified 63.4%

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

   if -3.2500000000000001e-81 < y < -6.5000000000000002e-258

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 59.9%

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

      1. unpow2 59.9%

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

   4. Simplified 59.9%

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

      1. associate-/l* 61.6%

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

      2. associate-/r/ 61.6%

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

   6. Applied egg-rr 61.6%

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

   if 9.79999999999999976e-43 < y < 1.0500000000000001e121

   1. Initial program 88.2%

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

   2. Taylor expanded in x around inf 50.0%

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

      1. unpow2 50.0%

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

   4. Simplified 50.0%

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

      1. associate-/r* 50.0%

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

      2. div-inv 50.0%

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

      3. associate-/l* 53.6%

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

      4. metadata-eval 53.6%

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

   6. Applied egg-rr 53.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.25 \cdot 10^{-81}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+161} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{+182}\right) \land x \cdot x \leq 10^{+243}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 2e+161)
         (and (not (<= (* x x) 4e+182)) (<= (* x x) 1e+243)))
   (* -0.5 (- (* z (/ z y)) y))
   (* (/ x (/ y x)) 0.5)))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e+161) || (!((x * x) <= 4e+182) && ((x * x) <= 1e+243))) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = (x / (y / x)) * 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 (((x * x) <= 2d+161) .or. (.not. ((x * x) <= 4d+182)) .and. ((x * x) <= 1d+243)) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (x / (y / x)) * 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 (((x * x) <= 2e+161) || (!((x * x) <= 4e+182) && ((x * x) <= 1e+243))) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = (x / (y / x)) * 0.5;
	}
	return tmp;
}

Python

z = abs(z)
def code(x, y, z):
	tmp = 0
	if ((x * x) <= 2e+161) or (not ((x * x) <= 4e+182) and ((x * x) <= 1e+243)):
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = (x / (y / x)) * 0.5
	return tmp

Julia

z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 2e+161) || (!(Float64(x * x) <= 4e+182) && (Float64(x * x) <= 1e+243)))
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(Float64(x / Float64(y / x)) * 0.5);
	end
	return tmp
end

MATLAB

z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 2e+161) || (~(((x * x) <= 4e+182)) && ((x * x) <= 1e+243)))
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = (x / (y / x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 2e+161], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 4e+182]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1e+243]]], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e161 or 4.0000000000000003e182 < (*.f64 x x) < 1.0000000000000001e243

   1. Initial program 72.4%

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

      1. sub-neg 72.4%

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

      2. +-commutative 72.4%

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

      3. neg-sub0 72.4%

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

      4. associate-+l- 72.4%

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

      5. sub0-neg 72.4%

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

      6. neg-mul-1 72.4%

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

      7. *-commutative 72.4%

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

      8. times-frac 72.4%

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

      9. associate--r+ 72.4%

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

      10. div-sub 72.4%

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

      11. difference-of-squares 72.4%

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

      12. +-commutative 72.4%

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

      13. associate-*l/ 76.0%

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

      14. *-commutative 76.0%

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

      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 99.8%

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

   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 z around inf 76.9%

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

      1. unpow2 76.9%

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

      2. associate-*r/ 83.2%

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

   8. Simplified 83.2%

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

   if 2.0000000000000001e161 < (*.f64 x x) < 4.0000000000000003e182 or 1.0000000000000001e243 < (*.f64 x x)

   1. Initial program 64.8%

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

   2. Taylor expanded in x around inf 70.3%

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

      1. unpow2 70.3%

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

   4. Simplified 70.3%

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

      1. associate-/r* 70.3%

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

      2. div-inv 70.3%

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

      3. associate-/l* 76.2%

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

      4. metadata-eval 76.2%

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

   6. Applied egg-rr 76.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+161} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{+182}\right) \land x \cdot x \leq 10^{+243}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \end{array} \]

Alternative 8: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-68)
   (* -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) <= 1e-68) {
		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) <= 1d-68) 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) <= 1e-68) {
		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) <= 1e-68:
		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) <= 1e-68)
		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) <= 1e-68)
		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], 1e-68], 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) < 1.00000000000000007e-68

   1. Initial program 70.6%

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

      1. sub-neg 70.6%

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

      2. +-commutative 70.6%

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

      3. neg-sub0 70.6%

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

      4. associate-+l- 70.6%

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

      5. sub0-neg 70.6%

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

      6. neg-mul-1 70.6%

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

      7. *-commutative 70.6%

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

      8. times-frac 70.6%

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

      9. associate--r+ 70.6%

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

      10. div-sub 70.6%

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

      11. difference-of-squares 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-*l/ 75.6%

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

      14. *-commutative 75.6%

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

      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 99.8%

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

   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 z around inf 82.7%

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

      1. unpow2 82.7%

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

      2. associate-*r/ 89.4%

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

   8. Simplified 89.4%

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

   if 1.00000000000000007e-68 < (*.f64 x x)

   1. Initial program 69.9%

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

      1. sub-neg 69.9%

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

      2. +-commutative 69.9%

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

      3. neg-sub0 69.9%

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

      4. associate-+l- 69.9%

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

      5. sub0-neg 69.9%

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

      6. neg-mul-1 69.9%

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

      7. *-commutative 69.9%

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

      8. times-frac 69.9%

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

      9. associate--r+ 69.9%

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

      10. div-sub 70.0%

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

      11. difference-of-squares 78.9%

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

      12. +-commutative 78.9%

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

      13. associate-*l/ 80.4%

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

      14. *-commutative 80.4%

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

      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. Taylor expanded in x around inf 86.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 9: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-68)
   (* -0.5 (- (/ (- z x) (/ y z)) 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) <= 1e-68) {
		tmp = -0.5 * (((z - x) / (y / z)) - 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) <= 1d-68) then
        tmp = (-0.5d0) * (((z - x) / (y / z)) - 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) <= 1e-68) {
		tmp = -0.5 * (((z - x) / (y / z)) - 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) <= 1e-68:
		tmp = -0.5 * (((z - x) / (y / z)) - 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) <= 1e-68)
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) / Float64(y / z)) - 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) <= 1e-68)
		tmp = -0.5 * (((z - x) / (y / z)) - 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], 1e-68], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] / N[(y / z), $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) < 1.00000000000000007e-68

   1. Initial program 70.6%

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

      1. sub-neg 70.6%

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

      2. +-commutative 70.6%

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

      3. neg-sub0 70.6%

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

      4. associate-+l- 70.6%

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

      5. sub0-neg 70.6%

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

      6. neg-mul-1 70.6%

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

      7. *-commutative 70.6%

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

      8. times-frac 70.6%

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

      9. associate--r+ 70.6%

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

      10. div-sub 70.6%

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

      11. difference-of-squares 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-*l/ 75.6%

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

      14. *-commutative 75.6%

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

      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 99.8%

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

   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 0 89.3%

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

   if 1.00000000000000007e-68 < (*.f64 x x)

   1. Initial program 69.9%

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

      1. sub-neg 69.9%

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

      2. +-commutative 69.9%

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

      3. neg-sub0 69.9%

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

      4. associate-+l- 69.9%

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

      5. sub0-neg 69.9%

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

      6. neg-mul-1 69.9%

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

      7. *-commutative 69.9%

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

      8. times-frac 69.9%

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

      9. associate--r+ 69.9%

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

      10. div-sub 70.0%

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

      11. difference-of-squares 78.9%

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

      12. +-commutative 78.9%

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

      13. associate-*l/ 80.4%

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

      14. *-commutative 80.4%

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

      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. Taylor expanded in x around inf 86.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-68}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 10: 51.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.22e-21)
   (* y 0.5)
   (if (<= y 5.5e+121) (* -0.5 (* z (/ z y))) (* y 0.5))))

C

z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.22e-21) {
		tmp = y * 0.5;
	} else if (y <= 5.5e+121) {
		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 <= (-1.22d-21)) then
        tmp = y * 0.5d0
    else if (y <= 5.5d+121) 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 <= -1.22e-21) {
		tmp = y * 0.5;
	} else if (y <= 5.5e+121) {
		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 <= -1.22e-21:
		tmp = y * 0.5
	elif y <= 5.5e+121:
		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 <= -1.22e-21)
		tmp = Float64(y * 0.5);
	elseif (y <= 5.5e+121)
		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 <= -1.22e-21)
		tmp = y * 0.5;
	elseif (y <= 5.5e+121)
		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, -1.22e-21], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 5.5e+121], 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 < -1.21999999999999991e-21 or 5.4999999999999998e121 < y

   1. Initial program 45.8%

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

   2. Taylor expanded in y around inf 68.0%

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

      1. *-commutative 68.0%

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

   4. Simplified 68.0%

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

   if -1.21999999999999991e-21 < y < 5.4999999999999998e121

   1. Initial program 87.6%

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

   2. Taylor expanded in y around 0 82.0%

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

      1. unpow2 82.0%

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

      2. unpow2 82.0%

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

   4. Simplified 82.0%

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

   5. Taylor expanded in x around 0 47.8%

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

      1. unpow2 47.8%

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

      2. associate-*l/ 49.3%

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

   7. Simplified 49.3%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-21}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 11: 34.0% 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 70.3%

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

2. Taylor expanded in y around inf 33.3%

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

   1. *-commutative 33.3%

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

4. Simplified 33.3%

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

5. Final simplification 33.3%

   \[\leadsto y \cdot 0.5 \]

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 2023240 
(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)))