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

Percentage Accurate: 70.1% → 99.9%

Time: 12.5s

Alternatives: 12

Speedup: 1.4×

• 42.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.5%

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

2. Taylor expanded in x around inf 74.5%

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

   1. distribute-lft-out 74.5%

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

   2. +-commutative 74.5%

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

   3. unpow2 74.5%

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

   4. unpow2 74.5%

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

   5. associate-/l* 81.9%

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

4. Simplified 81.9%

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

5. Taylor expanded in x around 0 74.5%

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

   1. associate--l+ 74.5%

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

   2. div-sub 80.3%

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

   3. unpow2 80.3%

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

   4. unpow2 80.3%

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

   5. difference-of-squares 86.6%

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

   6. associate-/l* 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 53.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e-154)
   (* z (* z (/ -0.5 y)))
   (if (<= (* x x) 5e-79)
     (* 0.5 y)
     (if (<= (* x x) 0.02)
       (* z (* (/ z y) -0.5))
       (if (<= (* x x) 1e+155) (* 0.5 y) (* x (/ x (* y 2.0))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = z * (z * (-0.5 / y));
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = z * ((z / y) * -0.5);
	} else if ((x * x) <= 1e+155) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d-154) then
        tmp = z * (z * ((-0.5d0) / y))
    else if ((x * x) <= 5d-79) then
        tmp = 0.5d0 * y
    else if ((x * x) <= 0.02d0) then
        tmp = z * ((z / y) * (-0.5d0))
    else if ((x * x) <= 1d+155) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x / (y * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = z * (z * (-0.5 / y));
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = z * ((z / y) * -0.5);
	} else if ((x * x) <= 1e+155) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e-154:
		tmp = z * (z * (-0.5 / y))
	elif (x * x) <= 5e-79:
		tmp = 0.5 * y
	elif (x * x) <= 0.02:
		tmp = z * ((z / y) * -0.5)
	elif (x * x) <= 1e+155:
		tmp = 0.5 * y
	else:
		tmp = x * (x / (y * 2.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e-154)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	elseif (Float64(x * x) <= 5e-79)
		tmp = Float64(0.5 * y);
	elseif (Float64(x * x) <= 0.02)
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	elseif (Float64(x * x) <= 1e+155)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e-154)
		tmp = z * (z * (-0.5 / y));
	elseif ((x * x) <= 5e-79)
		tmp = 0.5 * y;
	elseif ((x * x) <= 0.02)
		tmp = z * ((z / y) * -0.5);
	elseif ((x * x) <= 1e+155)
		tmp = 0.5 * y;
	else
		tmp = x * (x / (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-154], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-79], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 0.02], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+155], N[(0.5 * y), $MachinePrecision], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 5.0000000000000002e-154

   1. Initial program 68.9%

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

   2. Taylor expanded in z around inf 48.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot {z}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. mul-1-neg 48.9%

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

      2. unpow2 48.9%

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

      3. distribute-rgt-neg-in 48.9%

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

   4. Simplified 48.9%

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

      1. times-frac 55.9%

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

      2. metadata-eval 55.9%

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

      3. frac-2neg 55.9%

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

      4. div-inv 55.9%

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

      5. metadata-eval 55.9%

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

      6. *-commutative 55.9%

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

      7. associate-*r* 55.9%

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

      8. div-inv 55.9%

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

      9. associate-*l* 55.9%

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

      10. metadata-eval 55.9%

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

      11. div-inv 55.9%

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

      12. associate-/l/ 55.9%

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

      13. associate-/r* 55.9%

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

      14. metadata-eval 55.9%

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

   6. Applied egg-rr 55.9%

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

   if 5.0000000000000002e-154 < (*.f64 x x) < 4.99999999999999999e-79 or 0.0200000000000000004 < (*.f64 x x) < 1.00000000000000001e155

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 55.1%

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

   if 4.99999999999999999e-79 < (*.f64 x x) < 0.0200000000000000004

   1. Initial program 64.7%

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

   2. Taylor expanded in z around inf 51.4%

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

      1. associate-*r/ 51.4%

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

      2. unpow2 51.4%

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

      3. associate-*r* 51.4%

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

      4. *-commutative 51.4%

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

      5. metadata-eval 51.4%

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

      6. distribute-rgt-neg-in 51.4%

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

      7. associate-*r/ 57.0%

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

      8. distribute-rgt-neg-in 57.0%

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

      9. metadata-eval 57.0%

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

      10. associate-*l* 57.0%

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

   4. Simplified 57.0%

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

   if 1.00000000000000001e155 < (*.f64 x x)

   1. Initial program 65.3%

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

   2. Taylor expanded in x around inf 64.3%

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

      1. unpow2 64.3%

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

   4. Simplified 64.3%

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

      1. associate-/l* 69.0%

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

      2. associate-/r/ 69.0%

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

   6. Applied egg-rr 69.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 3: 53.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;\frac{\frac{z}{y}}{\frac{-2}{z}}\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e-154)
   (* z (* z (/ -0.5 y)))
   (if (<= (* x x) 5e-79)
     (* 0.5 y)
     (if (<= (* x x) 0.02)
       (/ (/ z y) (/ -2.0 z))
       (if (<= (* x x) 1e+155) (* 0.5 y) (* x (/ x (* y 2.0))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = z * (z * (-0.5 / y));
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = (z / y) / (-2.0 / z);
	} else if ((x * x) <= 1e+155) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d-154) then
        tmp = z * (z * ((-0.5d0) / y))
    else if ((x * x) <= 5d-79) then
        tmp = 0.5d0 * y
    else if ((x * x) <= 0.02d0) then
        tmp = (z / y) / ((-2.0d0) / z)
    else if ((x * x) <= 1d+155) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x / (y * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e-154) {
		tmp = z * (z * (-0.5 / y));
	} else if ((x * x) <= 5e-79) {
		tmp = 0.5 * y;
	} else if ((x * x) <= 0.02) {
		tmp = (z / y) / (-2.0 / z);
	} else if ((x * x) <= 1e+155) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e-154:
		tmp = z * (z * (-0.5 / y))
	elif (x * x) <= 5e-79:
		tmp = 0.5 * y
	elif (x * x) <= 0.02:
		tmp = (z / y) / (-2.0 / z)
	elif (x * x) <= 1e+155:
		tmp = 0.5 * y
	else:
		tmp = x * (x / (y * 2.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e-154)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	elseif (Float64(x * x) <= 5e-79)
		tmp = Float64(0.5 * y);
	elseif (Float64(x * x) <= 0.02)
		tmp = Float64(Float64(z / y) / Float64(-2.0 / z));
	elseif (Float64(x * x) <= 1e+155)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e-154)
		tmp = z * (z * (-0.5 / y));
	elseif ((x * x) <= 5e-79)
		tmp = 0.5 * y;
	elseif ((x * x) <= 0.02)
		tmp = (z / y) / (-2.0 / z);
	elseif ((x * x) <= 1e+155)
		tmp = 0.5 * y;
	else
		tmp = x * (x / (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-154], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-79], N[(0.5 * y), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 0.02], N[(N[(z / y), $MachinePrecision] / N[(-2.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+155], N[(0.5 * y), $MachinePrecision], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 5.0000000000000002e-154

   1. Initial program 68.9%

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

   2. Taylor expanded in z around inf 48.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot {z}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. mul-1-neg 48.9%

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

      2. unpow2 48.9%

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

      3. distribute-rgt-neg-in 48.9%

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

   4. Simplified 48.9%

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

      1. times-frac 55.9%

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

      2. metadata-eval 55.9%

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

      3. frac-2neg 55.9%

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

      4. div-inv 55.9%

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

      5. metadata-eval 55.9%

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

      6. *-commutative 55.9%

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

      7. associate-*r* 55.9%

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

      8. div-inv 55.9%

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

      9. associate-*l* 55.9%

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

      10. metadata-eval 55.9%

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

      11. div-inv 55.9%

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

      12. associate-/l/ 55.9%

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

      13. associate-/r* 55.9%

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

      14. metadata-eval 55.9%

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

   6. Applied egg-rr 55.9%

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

   if 5.0000000000000002e-154 < (*.f64 x x) < 4.99999999999999999e-79 or 0.0200000000000000004 < (*.f64 x x) < 1.00000000000000001e155

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 55.1%

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

   if 4.99999999999999999e-79 < (*.f64 x x) < 0.0200000000000000004

   1. Initial program 64.7%

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

   2. Taylor expanded in z around inf 51.4%

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

      1. *-commutative 51.4%

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

      2. unpow2 51.4%

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

      3. associate-/l* 56.9%

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

   4. Simplified 56.9%

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

      1. associate-/r/ 57.0%

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

      2. associate-*l* 57.0%

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

      3. metadata-eval 57.0%

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

      4. div-inv 57.0%

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

      5. frac-2neg 57.0%

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

      6. metadata-eval 57.0%

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

      7. associate-*r/ 57.0%

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

      8. associate-/l* 57.1%

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

      9. metadata-eval 57.1%

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

      10. frac-2neg 57.1%

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

   6. Applied egg-rr 57.1%

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

   if 1.00000000000000001e155 < (*.f64 x x)

   1. Initial program 65.3%

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

   2. Taylor expanded in x around inf 64.3%

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

      1. unpow2 64.3%

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

   4. Simplified 64.3%

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

      1. associate-/l* 69.0%

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

      2. associate-/r/ 69.0%

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

   6. Applied egg-rr 69.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-154}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \cdot x \leq 0.02:\\ \;\;\;\;\frac{\frac{z}{y}}{\frac{-2}{z}}\\ \mathbf{elif}\;x \cdot x \leq 10^{+155}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 4: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-9)
   (* 0.5 (+ y (/ x (/ y x))))
   (if (<= (* z z) 2e+307)
     (* (/ 0.5 y) (- (* x x) (* z z)))
     (* 0.5 (- y (* z (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else if ((z * z) <= 2e+307) {
		tmp = (0.5 / y) * ((x * x) - (z * z));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-9) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else if ((z * z) <= 2d+307) then
        tmp = (0.5d0 / y) * ((x * x) - (z * z))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else if ((z * z) <= 2e+307) {
		tmp = (0.5 / y) * ((x * x) - (z * z));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-9:
		tmp = 0.5 * (y + (x / (y / x)))
	elif (z * z) <= 2e+307:
		tmp = (0.5 / y) * ((x * x) - (z * z))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-9)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	elseif (Float64(z * z) <= 2e+307)
		tmp = Float64(Float64(0.5 / y) * Float64(Float64(x * x) - Float64(z * z)));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-9)
		tmp = 0.5 * (y + (x / (y / x)));
	elseif ((z * z) <= 2e+307)
		tmp = (0.5 / y) * ((x * x) - (z * z));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-9], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+307], N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.0000000000000001e-9

   1. Initial program 73.8%

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

   2. Taylor expanded in x around inf 89.7%

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

      1. distribute-lft-out 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

   5. Taylor expanded in x around 0 89.7%

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

      1. associate--l+ 89.7%

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

      2. div-sub 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. difference-of-squares 89.7%

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

      6. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 82.7%

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

      1. unpow2 82.7%

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

      2. associate-/l* 91.5%

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

   10. Simplified 91.5%

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

   if 5.0000000000000001e-9 < (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 83.4%

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

   2. Taylor expanded in y around 0 78.8%

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

      1. unpow2 78.8%

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

      2. unpow2 78.8%

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

   4. Simplified 78.8%

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

      1. div-inv 78.9%

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

      2. *-commutative 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-/r* 78.9%

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

      5. metadata-eval 78.9%

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

   6. Applied egg-rr 78.9%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 43.6%

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

   2. Taylor expanded in x around inf 44.1%

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

      1. distribute-lft-out 44.1%

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

      2. +-commutative 44.1%

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

      3. unpow2 44.1%

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

      4. unpow2 44.1%

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

      5. associate-/l* 68.2%

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

   4. Simplified 68.2%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. unpow2 64.9%

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

      2. associate-*r/ 89.0%

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

   7. Simplified 89.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + z}{\frac{y}{x - z}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-9)
   (* 0.5 (+ y (/ x (/ y x))))
   (/ (/ (+ x z) (/ y (- x z))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = ((x + z) / (y / (x - z))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-9) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = ((x + z) / (y / (x - z))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-9) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = ((x + z) / (y / (x - z))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-9:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = ((x + z) / (y / (x - z))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-9)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(Float64(Float64(x + z) / Float64(y / Float64(x - z))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-9)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = ((x + z) / (y / (x - z))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-9], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + z), $MachinePrecision] / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-9

   1. Initial program 73.8%

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

   2. Taylor expanded in x around inf 89.7%

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

      1. distribute-lft-out 89.7%

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

      2. +-commutative 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

   5. Taylor expanded in x around 0 89.7%

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

      1. associate--l+ 89.7%

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

      2. div-sub 89.7%

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

      3. unpow2 89.7%

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

      4. unpow2 89.7%

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

      5. difference-of-squares 89.7%

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

      6. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 82.7%

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

      1. unpow2 82.7%

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

      2. associate-/l* 91.5%

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

   10. Simplified 91.5%

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

   if 5.0000000000000001e-9 < (*.f64 z z)

   1. Initial program 61.2%

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

   2. Taylor expanded in y around 0 64.1%

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

      1. unpow2 64.1%

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

      2. unpow2 64.1%

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

   4. Simplified 64.1%

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

      1. add-exp-log_binary64 32.3%

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

   6. Applied rewrite-once 32.3%

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

      1. rem-exp-log 64.1%

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

      2. associate-/r* 64.1%

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

      3. difference-of-squares 76.6%

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

      4. associate-/l* 83.8%

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

   8. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{\frac{x + z}{\frac{y}{x - z}}}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + z}{\frac{y}{x - z}}}{2}\\ \end{array} \]

Alternative 6: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+234}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+234) (* 0.5 (+ y (* x (/ x y)))) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+234) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+234) then
        tmp = 0.5d0 * (y + (x * (x / y)))
    else
        tmp = z * ((z / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+234) {
		tmp = 0.5 * (y + (x * (x / y)));
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+234:
		tmp = 0.5 * (y + (x * (x / y)))
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+234)
		tmp = Float64(0.5 * Float64(y + Float64(x * Float64(x / y))));
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+234)
		tmp = 0.5 * (y + (x * (x / y)));
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+234], N[(0.5 * N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000004e234

   1. Initial program 74.7%

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

   2. Taylor expanded in x around inf 87.2%

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

      1. distribute-lft-out 87.2%

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

      2. +-commutative 87.2%

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

      3. unpow2 87.2%

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

      4. unpow2 87.2%

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

      5. associate-/l* 88.2%

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

   4. Simplified 88.2%

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

   5. Taylor expanded in z around 0 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*r/ 85.2%

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

   7. Simplified 85.2%

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

   if 2.00000000000000004e234 < (*.f64 z z)

   1. Initial program 54.3%

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

   2. Taylor expanded in z around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. unpow2 63.7%

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

      3. associate-*r* 63.7%

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

      4. *-commutative 63.7%

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

      5. metadata-eval 63.7%

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

      6. distribute-rgt-neg-in 63.7%

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

      7. associate-*r/ 71.7%

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

      8. distribute-rgt-neg-in 71.7%

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

      9. metadata-eval 71.7%

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

      10. associate-*l* 71.7%

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

   4. Simplified 71.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+234}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]

Alternative 7: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+180) (* 0.5 y) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+180) {
		tmp = 0.5 * y;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 4d+180) then
        tmp = 0.5d0 * y
    else
        tmp = z * ((z / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+180) {
		tmp = 0.5 * y;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+180:
		tmp = 0.5 * y
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+180)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e+180)
		tmp = 0.5 * y;
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+180], N[(0.5 * y), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4e180

   1. Initial program 74.1%

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

   2. Taylor expanded in y around inf 42.9%

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

   if 4e180 < (*.f64 z z)

   1. Initial program 57.4%

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

   2. Taylor expanded in z around inf 61.8%

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

      1. associate-*r/ 61.8%

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

      2. unpow2 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. metadata-eval 61.8%

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

      6. distribute-rgt-neg-in 61.8%

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

      7. associate-*r/ 68.9%

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

      8. distribute-rgt-neg-in 68.9%

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

      9. metadata-eval 68.9%

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

      10. associate-*l* 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+180}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]

Alternative 8: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.3e+120) (* 0.5 (+ y (/ x (/ y x)))) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.3e+120) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.3d+120) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = z * ((z / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.3e+120) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.3e+120:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.3e+120)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.3e+120)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.3e+120], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2999999999999999e120

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 80.4%

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

      1. distribute-lft-out 80.4%

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

      2. +-commutative 80.4%

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

      3. unpow2 80.4%

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

      4. unpow2 80.4%

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

      5. associate-/l* 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. associate--l+ 80.4%

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

      2. div-sub 85.7%

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

      3. unpow2 85.7%

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

      4. unpow2 85.7%

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

      5. difference-of-squares 88.6%

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

      6. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 66.3%

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

      1. unpow2 66.3%

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

      2. associate-/l* 73.0%

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

   10. Simplified 73.0%

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

   if 1.2999999999999999e120 < z

   1. Initial program 54.7%

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

   2. Taylor expanded in z around inf 61.9%

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

      1. associate-*r/ 61.9%

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

      2. unpow2 61.9%

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

      3. associate-*r* 61.9%

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

      4. *-commutative 61.9%

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

      5. metadata-eval 61.9%

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

      6. distribute-rgt-neg-in 61.9%

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

      7. associate-*r/ 71.3%

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

      8. distribute-rgt-neg-in 71.3%

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

      9. metadata-eval 71.3%

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

      10. associate-*l* 71.3%

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

   4. Simplified 71.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]

Alternative 9: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.06e+86) (* 0.5 (+ y (/ x (/ y x)))) (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.06e+86) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.06d+86) then
        tmp = 0.5d0 * (y + (x / (y / x)))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.06e+86) {
		tmp = 0.5 * (y + (x / (y / x)));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.06e+86:
		tmp = 0.5 * (y + (x / (y / x)))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.06e+86)
		tmp = Float64(0.5 * Float64(y + Float64(x / Float64(y / x))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.06e+86)
		tmp = 0.5 * (y + (x / (y / x)));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.06e+86], N[(0.5 * N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.06e86

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 80.3%

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

      1. distribute-lft-out 80.3%

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

      2. +-commutative 80.3%

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

      3. unpow2 80.3%

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

      4. unpow2 80.3%

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

      5. associate-/l* 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in x around 0 80.3%

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

      1. associate--l+ 80.3%

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

      2. div-sub 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. difference-of-squares 88.3%

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

      6. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 66.3%

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

      1. unpow2 66.3%

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

      2. associate-/l* 73.2%

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

   10. Simplified 73.2%

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

   if 1.06e86 < z

   1. Initial program 56.3%

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

   2. Taylor expanded in x around inf 52.6%

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

      1. distribute-lft-out 52.6%

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

      2. +-commutative 52.6%

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

      3. unpow2 52.6%

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

      4. unpow2 52.6%

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

      5. associate-/l* 70.5%

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

   4. Simplified 70.5%

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

   5. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*r/ 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 10: 39.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+304}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* x x) 1e+304) (* 0.5 y) (* x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+304) {
		tmp = 0.5 * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

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+304) then
        tmp = 0.5d0 * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+304) {
		tmp = 0.5 * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+304:
		tmp = 0.5 * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+304)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+304)
		tmp = 0.5 * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+304], N[(0.5 * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999994e303

   1. Initial program 70.7%

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

   2. Taylor expanded in y around inf 37.3%

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

   if 9.9999999999999994e303 < (*.f64 x x)

   1. Initial program 59.0%

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

   2. Taylor expanded in x around inf 69.1%

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

      1. unpow2 69.1%

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

   4. Simplified 69.1%

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

      1. associate-/l* 75.8%

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

      2. associate-/r/ 75.9%

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

   6. Applied egg-rr 75.9%

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

   8. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   9. Step-by-step derivation

      1. fma-udef 47.3%

         \[\leadsto \color{blue}{x \cdot x + 0} \]

      2. +-rgt-identity 47.3%

         \[\leadsto \color{blue}{x \cdot x} \]

   10. Simplified 47.3%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+304}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 11: 33.1% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 67.5%

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

2. Taylor expanded in y around inf 30.5%

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

3. Final simplification 30.5%

   \[\leadsto 0.5 \cdot y \]

Alternative 12: 2.9% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return 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 = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 67.5%

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

2. Taylor expanded in x around inf 33.1%

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

   1. unpow2 33.1%

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

4. Simplified 33.1%

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

   1. associate-/l* 36.1%

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

   2. associate-/r/ 36.1%

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

6. Applied egg-rr 36.1%

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

8. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{0 + x} \]
9. Step-by-step derivation

   1. +-lft-identity 2.8%

      \[\leadsto \color{blue}{x} \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{x} \]

11. Final simplification 2.8%

    \[\leadsto x \]

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 2023297 
(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)))