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

Percentage Accurate: 68.4% → 99.5%

Time: 7.4s

Alternatives: 3

Speedup: 36.3×

Specification

Math

\[\begin{array}{l} \\ x \cdot \sqrt{y \cdot y - z \cdot z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))

C

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

Java

public static double code(double x, double y, double z) {
	return x * Math.sqrt(((y * y) - (z * z)));
}

Python

def code(x, y, z):
	return x * math.sqrt(((y * y) - (z * z)))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{y \cdot y - z \cdot z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))

C

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

Java

public static double code(double x, double y, double z) {
	return x * Math.sqrt(((y * y) - (z * z)));
}

Python

def code(x, y, z):
	return x * math.sqrt(((y * y) - (z * z)))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (* x (* (sqrt (+ y_m z)) (sqrt (- y_m z)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x * (sqrt((y_m + z)) * sqrt((y_m - z)));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x * (sqrt((y_m + z)) * sqrt((y_m - z)))
end function

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x * (math.sqrt((y_m + z)) * math.sqrt((y_m - z)))

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x * N[(N[Sqrt[N[(y$95$m + z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(y$95$m - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x \cdot \sqrt{y \cdot y - z \cdot z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 67.8%

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

   2. difference-of-squares 69.5%

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

   3. unpow-prod-down 42.4%

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

4. Applied egg-rr 42.4%

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

   1. unpow1/2 42.4%

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

   2. unpow1/2 42.4%

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

6. Simplified 42.4%

   \[\leadsto x \cdot \color{blue}{\left(\sqrt{y + z} \cdot \sqrt{y - z}\right)} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 9.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (* x (+ y_m (* z (/ (* z -0.5) y_m)))))

C

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

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x * (y_m + (z * ((z * -0.5) / y_m)))

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x * N[(y$95$m + N[(z * N[(N[(z * -0.5), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x \cdot \sqrt{y \cdot y - z \cdot z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 67.8%

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

   2. difference-of-squares 69.5%

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

   3. unpow-prod-down 42.4%

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

4. Applied egg-rr 42.4%

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

   1. unpow1/2 42.4%

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

   2. unpow1/2 42.4%

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

6. Simplified 42.4%

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

7. Taylor expanded in z around 0 39.1%

   \[\leadsto \color{blue}{x \cdot y + z \cdot \left(-0.5 \cdot \frac{x \cdot \left(z \cdot \left(1 + 0.25 \cdot \frac{{\left(y + -1 \cdot y\right)}^{2}}{{y}^{2}}\right)\right)}{y} + 0.5 \cdot \frac{x \cdot \left(y + -1 \cdot y\right)}{y}\right)} \]

8. Taylor expanded in x around -inf 40.2%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y + z \cdot \left(-0.5 \cdot \frac{y + -1 \cdot y}{y} + 0.5 \cdot \frac{z \cdot \left(1 + 0.25 \cdot \frac{{\left(y + -1 \cdot y\right)}^{2}}{{y}^{2}}\right)}{y}\right)\right)\right)} \]

10. Simplified 43.8%

    \[\leadsto \color{blue}{\left(-x\right) \cdot \left(z \cdot \mathsf{fma}\left(0.5, \frac{z}{y}, 0\right) - y\right)} \]

11. Taylor expanded in x around 0 44.3%

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

    1. associate-*r* 44.3%

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

    2. sub-neg 44.3%

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

    3. distribute-lft-in 42.3%

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

    4. *-commutative 42.3%

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

    5. unpow2 42.3%

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

    6. associate-/l* 43.4%

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

    7. associate-*r* 43.4%

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

    8. *-commutative 43.4%

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

    9. distribute-lft-in 43.8%

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

    10. sub-neg 43.8%

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

    11. *-commutative 43.8%

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

    12. associate-*l* 43.8%

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

    13. neg-mul-1 43.8%

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

    14. sub-neg 43.8%

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

    15. +-commutative 43.8%

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

    16. distribute-neg-in 43.8%

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

13. Simplified 43.8%

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

Alternative 3: 99.3% accurate, 36.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot y\_m \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (* x y_m))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x * y_m;
}

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	return x * y_m

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[(x * y$95$m), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x \cdot \sqrt{y \cdot y - z \cdot z} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 43.4%

   \[\leadsto \color{blue}{x \cdot y} \]
4. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 2.5816096488251695e-278)
   (- (* x y))
   (* x (* (sqrt (+ y z)) (sqrt (- y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 2.5816096488251695e-278) {
		tmp = -(x * y);
	} else {
		tmp = x * (sqrt((y + z)) * sqrt((y - z)));
	}
	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 (y < 2.5816096488251695d-278) then
        tmp = -(x * y)
    else
        tmp = x * (sqrt((y + z)) * sqrt((y - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 2.5816096488251695e-278) {
		tmp = -(x * y);
	} else {
		tmp = x * (Math.sqrt((y + z)) * Math.sqrt((y - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 2.5816096488251695e-278:
		tmp = -(x * y)
	else:
		tmp = x * (math.sqrt((y + z)) * math.sqrt((y - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 2.5816096488251695e-278)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(x * Float64(sqrt(Float64(y + z)) * sqrt(Float64(y - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 2.5816096488251695e-278)
		tmp = -(x * y);
	else
		tmp = x * (sqrt((y + z)) * sqrt((y - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 2.5816096488251695e-278], (-N[(x * y), $MachinePrecision]), N[(x * N[(N[Sqrt[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(y - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024191 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 5163219297650339/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z))))))

  (* x (sqrt (- (* y y) (* z z)))))