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

Percentage Accurate: 68.1% → 99.4%

Time: 8.6s

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: 68.1% 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: 99.4% accurate, 0.5× speedup

Math

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

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(Float64(x * 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[(N[(x * N[Sqrt[N[(y$95$m + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(y$95$m - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

   1. pow1/2 N/A

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

   2. difference-of-squares N/A

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

   3. unpow-prod-down N/A

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

   4. associate-*r* N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {\left(y + z\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(y - z\right)}^{\frac{1}{2}}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({\left(y + z\right)}^{\frac{1}{2}}\right)\right), \left({\color{blue}{\left(y - z\right)}}^{\frac{1}{2}}\right)\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\left(y + z\right), \frac{1}{2}\right)\right), \left({\left(y - \color{blue}{z}\right)}^{\frac{1}{2}}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \left({\left(y - z\right)}^{\frac{1}{2}}\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \mathsf{pow.f64}\left(\left(y - z\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   10. --lowering--.f64 48.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(y, z\right), \frac{1}{2}\right)\right) \]

4. Applied egg-rr 48.9%

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

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \left(\sqrt{y - z}\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\left(y - z\right)\right)\right) \]

   3. --lowering--.f64 48.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(y, z\right), \frac{1}{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(y, z\right)\right)\right) \]

6. Applied egg-rr 48.9%

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

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\sqrt{y + z}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\left(y + z\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   3. +-lowering-+.f64 48.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(y, z\right)\right)\right) \]

8. Applied egg-rr 48.9%

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

Alternative 2: 99.7% accurate, 9.9× speedup

Math

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

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(z * Float64(-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[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. associate-*r/ N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

      \[\leadsto x \cdot \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{y} \cdot {z}^{2} + y\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{y}\right)\right) \cdot {z}^{2} + y\right) \]

   10. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{y}\right)\right) \cdot {z}^{2} + y\right) \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{y}\right)\right) \cdot {z}^{2} + y\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{y}\right)\right) \cdot {z}^{2} + y\right)}\right) \]

5. Simplified 50.2%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \left(\frac{\frac{-1}{2} \cdot \left(z \cdot z\right)}{y}\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \left(\frac{\frac{-1}{2}}{y} \cdot \color{blue}{\left(z \cdot z\right)}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \left(\left(\frac{\frac{-1}{2}}{y} \cdot z\right) \cdot \color{blue}{z}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{y} \cdot z\right), \color{blue}{z}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{y}\right), z\right), z\right)\right)\right) \]

   6. /-lowering-/.f64 51.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, y\right), z\right), z\right)\right)\right) \]

7. Applied egg-rr 51.8%

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

8. Final simplification 51.8%

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

Alternative 3: 99.4% 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 70.5%

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

3. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 51.4%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

5. Simplified 51.4%

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

Developer Target 1: 99.4% 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 2024145 
(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)))))