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

Percentage Accurate: 68.4% → 99.9%

Time: 6.3s

Alternatives: 7

Speedup: 1.3×

• 41.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + x, \frac{x - z}{y}, y\right) \cdot 0.5} \]
6. Add Preprocessing

Alternative 2: 38.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -2e-125)
     (* (* (/ -0.5 y) z) z)
     (if (<= t_0 5e+152) (* 0.5 y) (* (* (/ x y) x) 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-125) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= 5e+152) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 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) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-2d-125)) then
        tmp = (((-0.5d0) / y) * z) * z
    else if (t_0 <= 5d+152) then
        tmp = 0.5d0 * y
    else
        tmp = ((x / y) * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-125) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= 5e+152) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -2e-125:
		tmp = ((-0.5 / y) * z) * z
	elif t_0 <= 5e+152:
		tmp = 0.5 * y
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-125)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	elseif (t_0 <= 5e+152)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-125)
		tmp = ((-0.5 / y) * z) * z;
	elseif (t_0 <= 5e+152)
		tmp = 0.5 * y;
	else
		tmp = ((x / y) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-125], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+152], N[(0.5 * y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.00000000000000002e-125

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 25.5%

      \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.5%

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

   if -2.00000000000000002e-125 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5e152

   1. Initial program 80.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if 5e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 58.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 36.3%

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

Alternative 3: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\left(y - \frac{z}{y} \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x - z}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -2e-125)
   (* (- y (* (/ z y) z)) 0.5)
   (* (fma x (/ (- x z) y) y) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125) {
		tmp = (y - ((z / y) * z)) * 0.5;
	} else {
		tmp = fma(x, ((x - z) / y), y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -2e-125)
		tmp = Float64(Float64(y - Float64(Float64(z / y) * z)) * 0.5);
	else
		tmp = Float64(fma(x, Float64(Float64(x - z) / y), y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-125], N[(N[(y - N[(N[(z / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.00000000000000002e-125

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 60.6%

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

   if -2.00000000000000002e-125 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 56.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, z\right), \frac{x - z}{y}, y\right) \cdot 0.5 \]

   8. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(x \cdot {\left(\sqrt{-1}\right)}^{2}\right), \frac{x - z}{y}, y\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(x, \frac{x - z}{y}, y\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\left(y - \frac{z}{y} \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -2e-125)
   (* (- y (* (/ z y) z)) 0.5)
   (* (fma (/ x y) x y) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125) {
		tmp = (y - ((z / y) * z)) * 0.5;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -2e-125)
		tmp = Float64(Float64(y - Float64(Float64(z / y) * z)) * 0.5);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-125], N[(N[(y - N[(N[(z / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.00000000000000002e-125

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 60.6%

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

   if -2.00000000000000002e-125 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, y\right)} \cdot \frac{1}{2} \]

      18. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

Alternative 5: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -2e-125)
   (* (* (/ -0.5 y) z) z)
   (* (fma (/ x y) x y) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125) {
		tmp = ((-0.5 / y) * z) * z;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -2e-125)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-125], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.00000000000000002e-125

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 25.5%

      \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.5%

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

   if -2.00000000000000002e-125 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, y\right)} \cdot \frac{1}{2} \]

      18. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

Alternative 6: 34.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -2e-125)
   (* (* (/ -0.5 y) z) z)
   (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125) {
		tmp = ((-0.5 / y) * z) * z;
	} else {
		tmp = 0.5 * 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 (((((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)) <= (-2d-125)) then
        tmp = (((-0.5d0) / y) * z) * z
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125) {
		tmp = ((-0.5 / y) * z) * z;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125:
		tmp = ((-0.5 / y) * z) * z
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -2e-125)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -2e-125)
		tmp = ((-0.5 / y) * z) * z;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-125], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.00000000000000002e-125

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 25.5%

      \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.5%

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

   if -2.00000000000000002e-125 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 30.9

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

   5. Applied rewrites 30.9%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 34.9% accurate, 6.3× 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 70.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 33.3

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

5. Applied rewrites 33.3%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.1× 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 2024329 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))