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

Percentage Accurate: 67.1% → 99.5%

Time: 4.1s

Alternatives: 4

Speedup: 18.0×

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: 67.1% 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: 99.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \left(0.5 \cdot t_0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + t_0 \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (/ y z))))
   (if (<= y -1e-262) (* x (- (* 0.5 t_0) y)) (* x (+ y (* t_0 -0.5))))))

C

double code(double x, double y, double z) {
	double t_0 = z / (y / z);
	double tmp;
	if (y <= -1e-262) {
		tmp = x * ((0.5 * t_0) - y);
	} else {
		tmp = x * (y + (t_0 * -0.5));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = z / (y / z)
    if (y <= (-1d-262)) then
        tmp = x * ((0.5d0 * t_0) - y)
    else
        tmp = x * (y + (t_0 * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z / (y / z);
	double tmp;
	if (y <= -1e-262) {
		tmp = x * ((0.5 * t_0) - y);
	} else {
		tmp = x * (y + (t_0 * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z / (y / z)
	tmp = 0
	if y <= -1e-262:
		tmp = x * ((0.5 * t_0) - y)
	else:
		tmp = x * (y + (t_0 * -0.5))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z / Float64(y / z))
	tmp = 0.0
	if (y <= -1e-262)
		tmp = Float64(x * Float64(Float64(0.5 * t_0) - y));
	else
		tmp = Float64(x * Float64(y + Float64(t_0 * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z / (y / z);
	tmp = 0.0;
	if (y <= -1e-262)
		tmp = x * ((0.5 * t_0) - y);
	else
		tmp = x * (y + (t_0 * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-262], N[(x * N[(N[(0.5 * t$95$0), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000001e-262

   1. Initial program 67.8%

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

   2. Taylor expanded in y around -inf 92.8%

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

      1. mul-1-neg 92.8%

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

      2. unsub-neg 92.8%

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

      3. unpow2 92.8%

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

      4. associate-/l* 99.8%

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

   4. Simplified 99.8%

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

   if -1.00000000000000001e-262 < y

   1. Initial program 63.4%

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

   2. Taylor expanded in y around inf 92.1%

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

      1. unpow2 92.1%

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

   4. Simplified 92.1%

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

   5. Taylor expanded in z around 0 92.1%

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

      1. *-commutative 92.1%

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

      2. unpow2 92.1%

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

      3. associate-/l* 99.5%

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

   7. Simplified 99.5%

      \[\leadsto x \cdot \left(y + \color{blue}{\frac{z}{\frac{y}{z}} \cdot -0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z}{\frac{y}{z}} \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z}{\frac{y}{z}} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-262) (* y (- x)) (* x (+ y (* (/ z (/ y z)) -0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-262) {
		tmp = y * -x;
	} else {
		tmp = x * (y + ((z / (y / z)) * -0.5));
	}
	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 <= (-1d-262)) then
        tmp = y * -x
    else
        tmp = x * (y + ((z / (y / z)) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-262) {
		tmp = y * -x;
	} else {
		tmp = x * (y + ((z / (y / z)) * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-262:
		tmp = y * -x
	else:
		tmp = x * (y + ((z / (y / z)) * -0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-262)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(x * Float64(y + Float64(Float64(z / Float64(y / z)) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-262)
		tmp = y * -x;
	else
		tmp = x * (y + ((z / (y / z)) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-262], N[(y * (-x)), $MachinePrecision], N[(x * N[(y + N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000001e-262

   1. Initial program 67.8%

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

   2. Taylor expanded in y around -inf 99.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.6%

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

      2. mul-1-neg 99.6%

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

   4. Simplified 99.6%

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

   if -1.00000000000000001e-262 < y

   1. Initial program 63.4%

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

   2. Taylor expanded in y around inf 92.1%

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

      1. unpow2 92.1%

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

   4. Simplified 92.1%

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

   5. Taylor expanded in z around 0 92.1%

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

      1. *-commutative 92.1%

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

      2. unpow2 92.1%

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

      3. associate-/l* 99.5%

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

   7. Simplified 99.5%

      \[\leadsto x \cdot \left(y + \color{blue}{\frac{z}{\frac{y}{z}} \cdot -0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z}{\frac{y}{z}} \cdot -0.5\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y -1e-262) (* y (- x)) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-262) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	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 <= (-1d-262)) then
        tmp = y * -x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-262) {
		tmp = y * -x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-262:
		tmp = y * -x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-262)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-262)
		tmp = y * -x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-262], N[(y * (-x)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000001e-262

   1. Initial program 67.8%

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

   2. Taylor expanded in y around -inf 99.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.6%

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

      2. mul-1-neg 99.6%

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

   4. Simplified 99.6%

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

   if -1.00000000000000001e-262 < y

   1. Initial program 63.4%

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

   2. Taylor expanded in y around inf 99.4%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 50.6% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return y * x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in y around inf 51.9%

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

3. Final simplification 51.9%

   \[\leadsto y \cdot x \]

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 2023196 
(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)))))