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

Average Error: 38.88% → 0.63%

Time: 5.5s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y - z} \cdot \sqrt{y + z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-289)
   (* x (fma 0.5 (/ z (/ y z)) (- y)))
   (* x (* (sqrt (- y z)) (sqrt (+ 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 <= -5e-289) {
		tmp = x * fma(0.5, (z / (y / z)), -y);
	} else {
		tmp = x * (sqrt((y - z)) * sqrt((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 <= -5e-289)
		tmp = Float64(x * fma(0.5, Float64(z / Float64(y / z)), Float64(-y)));
	else
		tmp = Float64(x * Float64(sqrt(Float64(y - z)) * sqrt(Float64(y + z))));
	end
	return 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, -5e-289], N[(x * N[(0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sqrt[N[(y - z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	38.88%
Target	0.86%
Herbie	0.63%

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

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000029e-289

   1. Initial program 39.08

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

   2. Taylor expanded in y around -inf 5.31

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

   3. Simplified 0.46

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

      [Start] 5.31	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      fma-def [=>] 5.31	\[ x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{z}^{2}}{y}, -1 \cdot y\right)} \]
      unpow2 [=>] 5.31	\[ x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{z \cdot z}}{y}, -1 \cdot y\right) \]
      associate-/l* [=>] 0.46	\[ x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{z}{\frac{y}{z}}}, -1 \cdot y\right) \]
      mul-1-neg [=>] 0.46	\[ x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, \color{blue}{-y}\right) \]

   if -5.00000000000000029e-289 < y

   1. Initial program 38.69

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

   2. Applied egg-rr 0.8

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

4. Final simplification 0.63

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y - z} \cdot \sqrt{y + z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.46%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 2
Error	0.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 3
Error	0.94%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	47.1%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023090 
(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)))))