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

Average Error: 24.7 → 0.7

Time: 3.2s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4215422919260029 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(z \cdot \frac{z}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4215422919260029e-252)
   (* x (- (* 0.5 (* z (/ z y))) y))
   (* y x)))

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 <= -1.4215422919260029e-252) {
		tmp = x * ((0.5 * (z * (z / y))) - y);
	} 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
    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 <= (-1.4215422919260029d-252)) then
        tmp = x * ((0.5d0 * (z * (z / y))) - y)
    else
        tmp = y * x
    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 <= -1.4215422919260029e-252) {
		tmp = x * ((0.5 * (z * (z / y))) - y);
	} else {
		tmp = y * x;
	}
	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 <= -1.4215422919260029e-252:
		tmp = x * ((0.5 * (z * (z / y))) - y)
	else:
		tmp = y * x
	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 <= -1.4215422919260029e-252)
		tmp = Float64(x * Float64(Float64(0.5 * Float64(z * Float64(z / y))) - y));
	else
		tmp = Float64(y * x);
	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 <= -1.4215422919260029e-252)
		tmp = x * ((0.5 * (z * (z / y))) - y);
	else
		tmp = y * x;
	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, -1.4215422919260029e-252], N[(x * N[(N[(0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Target

Original	24.7
Target	0.6
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if y < -1.4215422919260029e-252

   1. Initial program 24.8

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

   2. Taylor expanded in y around -inf 3.7

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

   3. Simplified 3.7

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

      [Start] 3.7	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      mul-1-neg [=>] 3.7	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + \color{blue}{\left(-y\right)}\right) \]
      unsub-neg [=>] 3.7	\[ x \cdot \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} - y\right)} \]
      associate-*r/ [=>] 3.7	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot {z}^{2}}{y}} - y\right) \]
      unpow2 [=>] 3.7	\[ x \cdot \left(\frac{0.5 \cdot \color{blue}{\left(z \cdot z\right)}}{y} - y\right) \]

   4. Taylor expanded in z around 0 3.7

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

   5. Simplified 0.4

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

      [Start] 3.7	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right) \]
      associate-*r/ [=>] 3.7	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot {z}^{2}}{y}} - y\right) \]
      unpow2 [=>] 3.7	\[ x \cdot \left(\frac{0.5 \cdot \color{blue}{\left(z \cdot z\right)}}{y} - y\right) \]
      associate-*r* [=>] 3.7	\[ x \cdot \left(\frac{\color{blue}{\left(0.5 \cdot z\right) \cdot z}}{y} - y\right) \]
      associate-*r/ [<=] 0.4	\[ x \cdot \left(\color{blue}{\left(0.5 \cdot z\right) \cdot \frac{z}{y}} - y\right) \]
      associate-*l* [=>] 0.4	\[ x \cdot \left(\color{blue}{0.5 \cdot \left(z \cdot \frac{z}{y}\right)} - y\right) \]

   if -1.4215422919260029e-252 < y

   1. Initial program 24.7

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

   2. Taylor expanded in y around inf 0.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4215422919260029 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(z \cdot \frac{z}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4215422919260029 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	29.6
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023018 
(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)))))