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

Percentage Accurate: 67.2% → 99.4%

Time: 17.7s

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: 67.2% 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

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (* x (* (sqrt (+ z y)) (sqrt (- y z)))))

C

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

Fortran

NOTE: y should be positive before calling this function
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((z + y)) * sqrt((y - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x * N[(N[Sqrt[N[(z + y), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(y - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

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

   1. difference-of-squares 71.9%

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

   2. sqrt-prod 45.5%

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

3. Applied egg-rr 45.5%

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

   1. +-commutative 45.5%

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

5. Simplified 45.5%

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

6. Final simplification 45.5%

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

Alternative 2: 99.6% accurate, 9.9× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (* x (+ y (/ (* z -0.5) (/ y z)))))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (y + ((z * (-0.5d0)) / (y / z)))
end function

Java

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

Python

y = abs(y)
def code(x, y, z):
	return x * (y + ((z * -0.5) / (y / z)))

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x * N[(y + N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

2. Taylor expanded in y around inf 48.0%

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

   1. unpow2 48.0%

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

4. Simplified 48.0%

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

   1. associate-/l* 48.3%

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

   2. *-commutative 48.3%

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

   3. associate-*l/ 48.3%

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

6. Applied egg-rr 48.3%

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

7. Final simplification 48.3%

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

Alternative 3: 99.3% accurate, 36.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ x \cdot y \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (* x y))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * y
end function

Java

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

Python

y = abs(y)
def code(x, y, z):
	return x * y

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

2. Taylor expanded in y around inf 48.1%

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

3. Final simplification 48.1%

   \[\leadsto x \cdot y \]

Developer target: 99.3% 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 2023278 
(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)))))