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

Percentage Accurate: 68.4% → 99.6%

Time: 5.7s

Alternatives: 2

Speedup: 4.8×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 55.8

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

8. Applied rewrites 55.8%

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

10. Applied rewrites 56.5%

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

12. Applied rewrites 58.5%

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

Alternative 2: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

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 = y_m * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 58.3

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

5. Applied rewrites 58.3%

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

Developer Target 1: 99.3% accurate, 0.7× 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 2024326 
(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)))))