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

Average Error: 25.2 → 0.4

Time: 6.9s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \left(z \cdot \left(z \cdot \frac{0.5}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.6e-262)
   (* x (- (* z (* z (/ 0.5 y))) y))
   (* x (+ y (/ (* z -0.5) (/ y z))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e-262) {
		tmp = x * ((z * (z * (0.5 / y))) - y);
	} else {
		tmp = x * (y + ((z * -0.5) / (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
    code = x * sqrt(((y * y) - (z * z)))
end function

↓

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 <= (-3.6d-262)) then
        tmp = x * ((z * (z * (0.5d0 / y))) - y)
    else
        tmp = x * (y + ((z * (-0.5d0)) / (y / z)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.6e-262) {
		tmp = x * ((z * (z * (0.5 / y))) - y);
	} else {
		tmp = x * (y + ((z * -0.5) / (y / z)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -3.6e-262:
		tmp = x * ((z * (z * (0.5 / y))) - y)
	else:
		tmp = x * (y + ((z * -0.5) / (y / z)))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.6e-262)
		tmp = Float64(x * Float64(Float64(z * Float64(z * Float64(0.5 / y))) - y));
	else
		tmp = Float64(x * Float64(y + Float64(Float64(z * -0.5) / Float64(y / z))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.6e-262)
		tmp = x * ((z * (z * (0.5 / y))) - y);
	else
		tmp = x * (y + ((z * -0.5) / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, -3.6e-262], N[(x * N[(N[(z * N[(z * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	25.2
Target	0.6
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if y < -3.5999999999999998e-262

   1. Initial program 25.5

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

   2. Taylor expanded in y around -inf 3.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right)} \]

   3. Simplified 2.1

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

      [Start] 3.6	\[ 0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right) \]
      mul-1-neg [=>] 3.6	\[ 0.5 \cdot \frac{{z}^{2} \cdot x}{y} + \color{blue}{\left(-y \cdot x\right)} \]
      unsub-neg [=>] 3.6	\[ \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} - y \cdot x} \]
      associate-*r/ [=>] 3.6	\[ \color{blue}{\frac{0.5 \cdot \left({z}^{2} \cdot x\right)}{y}} - y \cdot x \]
      associate-/l* [=>] 3.6	\[ \color{blue}{\frac{0.5}{\frac{y}{{z}^{2} \cdot x}}} - y \cdot x \]
      unpow2 [=>] 3.6	\[ \frac{0.5}{\frac{y}{\color{blue}{\left(z \cdot z\right)} \cdot x}} - y \cdot x \]
      associate-*l* [=>] 2.1	\[ \frac{0.5}{\frac{y}{\color{blue}{z \cdot \left(z \cdot x\right)}}} - y \cdot x \]

   4. Taylor expanded in y around 0 3.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right)} \]

   5. Simplified 0.3

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

      [Start] 3.6	\[ 0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right) \]
      associate-*r/ [=>] 3.6	\[ \color{blue}{\frac{0.5 \cdot \left({z}^{2} \cdot x\right)}{y}} + -1 \cdot \left(y \cdot x\right) \]
      unpow2 [=>] 3.6	\[ \frac{0.5 \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)}{y} + -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 3.6	\[ \frac{\color{blue}{\left(0.5 \cdot \left(z \cdot z\right)\right) \cdot x}}{y} + -1 \cdot \left(y \cdot x\right) \]
      associate-*l/ [<=] 3.2	\[ \color{blue}{\frac{0.5 \cdot \left(z \cdot z\right)}{y} \cdot x} + -1 \cdot \left(y \cdot x\right) \]
      associate-*l/ [<=] 3.2	\[ \color{blue}{\left(\frac{0.5}{y} \cdot \left(z \cdot z\right)\right)} \cdot x + -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 3.2	\[ \left(\frac{0.5}{y} \cdot \left(z \cdot z\right)\right) \cdot x + \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 3.2	\[ \left(\frac{0.5}{y} \cdot \left(z \cdot z\right)\right) \cdot x + \color{blue}{\left(-y\right)} \cdot x \]
      distribute-rgt-in [<=] 3.2	\[ \color{blue}{x \cdot \left(\frac{0.5}{y} \cdot \left(z \cdot z\right) + \left(-y\right)\right)} \]
      associate-*l/ [=>] 3.2	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot \left(z \cdot z\right)}{y}} + \left(-y\right)\right) \]
      unpow2 [<=] 3.2	\[ x \cdot \left(\frac{0.5 \cdot \color{blue}{{z}^{2}}}{y} + \left(-y\right)\right) \]
      associate-*r/ [<=] 3.2	\[ x \cdot \left(\color{blue}{0.5 \cdot \frac{{z}^{2}}{y}} + \left(-y\right)\right) \]
      sub-neg [<=] 3.2	\[ x \cdot \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} - y\right)} \]
      associate-*r/ [=>] 3.2	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot {z}^{2}}{y}} - y\right) \]
      unpow2 [=>] 3.2	\[ x \cdot \left(\frac{0.5 \cdot \color{blue}{\left(z \cdot z\right)}}{y} - y\right) \]
      associate-*l/ [<=] 3.2	\[ x \cdot \left(\color{blue}{\frac{0.5}{y} \cdot \left(z \cdot z\right)} - y\right) \]
      *-commutative [=>] 3.2	\[ x \cdot \left(\color{blue}{\left(z \cdot z\right) \cdot \frac{0.5}{y}} - y\right) \]
      associate-*l* [=>] 0.3	\[ x \cdot \left(\color{blue}{z \cdot \left(z \cdot \frac{0.5}{y}\right)} - y\right) \]

   if -3.5999999999999998e-262 < y

   1. Initial program 24.9

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

   2. Taylor expanded in y around inf 3.4

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

   3. Simplified 3.4

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

      [Start] 3.4	\[ x \cdot \left(y + -0.5 \cdot \frac{{z}^{2}}{y}\right) \]
      unpow2 [=>] 3.4	\[ x \cdot \left(y + -0.5 \cdot \frac{\color{blue}{z \cdot z}}{y}\right) \]

   4. Applied egg-rr 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \left(z \cdot \left(z \cdot \frac{0.5}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	30.3
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023031 
(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)))))