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

Percentage Accurate: 67.9% → 99.4%
Time: 5.0s
Alternatives: 2
Speedup: 36.3×

Specification

?
\[\begin{array}{l} \\ x \cdot \sqrt{y \cdot y - z \cdot z} \end{array} \]
(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}

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

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

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

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

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

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

\begin{array}{l}

\\
x \cdot \sqrt{y \cdot y - z \cdot z}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 67.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot \sqrt{y \cdot y - z \cdot z} \end{array} \]
(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}

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

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

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

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

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

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

\begin{array}{l}

\\
x \cdot \sqrt{y \cdot y - z \cdot z}
\end{array}

Alternative 1: 99.4% accurate, 0.5× speedup?

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \left(\sqrt{z + y\_m} \cdot \sqrt{y\_m - z}\right) \end{array} \]
y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (* x (* (sqrt (+ z y_m)) (sqrt (- y_m z)))))
y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x * (sqrt((z + y_m)) * sqrt((y_m - z)));
}

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((z + y_m)) * sqrt((y_m - z)))
end function

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

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

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

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

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

\begin{array}{l}
y_m = \left|y\right|

\\
x \cdot \left(\sqrt{z + y\_m} \cdot \sqrt{y\_m - z}\right)
\end{array}
Derivation

  1. Initial program 68.6%

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

    1. pow1/268.6%

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

    2. difference-of-squares71.0%

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

    3. unpow-prod-down52.4%

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

  4. Applied egg-rr52.4%

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

    1. unpow1/252.4%

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

    2. unpow1/252.4%

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

    3. +-commutative52.4%

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

  6. Simplified52.4%

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

  7. Final simplification52.4%

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

Alternative 2: 99.1% accurate, 36.3× speedup?

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot y\_m \end{array} \]
y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (* x y_m))
y_m = fabs(y);
double code(double x, double y_m, double z) {
	return x * y_m;
}

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

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

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

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

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

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

\begin{array}{l}
y_m = \left|y\right|

\\
x \cdot y\_m
\end{array}
Derivation

  1. Initial program 68.6%

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

  3. Taylor expanded in y around inf 54.4%

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

    1. *-commutative54.4%

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

  5. Simplified54.4%

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

  6. Final simplification54.4%

    \[\leadsto x \cdot y \]
  7. Add Preprocessing

Developer target: 99.4% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \end{array} \]
(FPCore (x y z)
 :precision binary64
 (if (< y 2.5816096488251695e-278)
   (- (* x y))
   (* x (* (sqrt (+ y z)) (sqrt (- y z))))))
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;
}

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

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;
}

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

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

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

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]]

\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}

Reproduce

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herbie shell --seed 2024044 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

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