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

Average Accuracy: 61.1% → 99.2%

Time: 9.0s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot \left(x \cdot \frac{z}{\frac{y}{z}}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e-264) (* y (- x)) (fma y x (* -0.5 (* x (/ z (/ 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 <= -2.3e-264) {
		tmp = y * -x;
	} else {
		tmp = fma(y, x, (-0.5 * (x * (z / (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 <= -2.3e-264)
		tmp = Float64(y * Float64(-x));
	else
		tmp = fma(y, x, Float64(-0.5 * Float64(x * Float64(z / 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, -2.3e-264], N[(y * (-x)), $MachinePrecision], N[(y * x + N[(-0.5 * N[(x * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	61.1%
Target	99.1%
Herbie	99.2%

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

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000012e-264

   1. Initial program 61.2%

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

   2. Taylor expanded in y around -inf 99.1%

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

   3. Simplified 99.1%

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

      [Start] 99.1	\[ -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 99.1	\[ \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 99.1	\[ \color{blue}{\left(-y\right)} \cdot x \]

   if -2.30000000000000012e-264 < y

   1. Initial program 61.0%

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

   2. Taylor expanded in y around inf 94.2%

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

   3. Simplified 99.3%

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

      [Start] 94.2	\[ -0.5 \cdot \frac{{z}^{2} \cdot x}{y} + y \cdot x \]
      +-commutative [=>] 94.2	\[ \color{blue}{y \cdot x + -0.5 \cdot \frac{{z}^{2} \cdot x}{y}} \]
      fma-def [=>] 94.2	\[ \color{blue}{\mathsf{fma}\left(y, x, -0.5 \cdot \frac{{z}^{2} \cdot x}{y}\right)} \]
      associate-/l* [=>] 90.1	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \color{blue}{\frac{{z}^{2}}{\frac{y}{x}}}\right) \]
      unpow2 [=>] 90.1	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \frac{\color{blue}{z \cdot z}}{\frac{y}{x}}\right) \]
      associate-/r/ [=>] 94.9	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \color{blue}{\left(\frac{z \cdot z}{y} \cdot x\right)}\right) \]
      associate-/l* [=>] 99.3	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \left(\color{blue}{\frac{z}{\frac{y}{z}}} \cdot x\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot \left(x \cdot \frac{z}{\frac{y}{z}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{-0.5 \cdot z}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-264}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	53.4%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023141 
(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)))))