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

Average Accuracy: 61.8% → 99.6%

Time: 9.9s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_0 := \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(0.5 \cdot t_0 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot \left(t_0 \cdot x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (/ y z))))
   (if (<= y -2e-304) (* (- (* 0.5 t_0) y) x) (fma y x (* -0.5 (* t_0 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 t_0 = z / (y / z);
	double tmp;
	if (y <= -2e-304) {
		tmp = ((0.5 * t_0) - y) * x;
	} else {
		tmp = fma(y, x, (-0.5 * (t_0 * 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)
	t_0 = Float64(z / Float64(y / z))
	tmp = 0.0
	if (y <= -2e-304)
		tmp = Float64(Float64(Float64(0.5 * t_0) - y) * x);
	else
		tmp = fma(y, x, Float64(-0.5 * Float64(t_0 * x)));
	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_] := Block[{t$95$0 = N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e-304], N[(N[(N[(0.5 * t$95$0), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision], N[(y * x + N[(-0.5 * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	61.8%
Target	99.2%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999994e-304

   1. Initial program 61.8%

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

   2. Taylor expanded in y around -inf 95.8%

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

   3. Simplified 99.7%

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

      [Start] 95.8	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      fma-def [=>] 95.8	\[ x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{z}^{2}}{y}, -1 \cdot y\right)} \]
      unpow2 [=>] 95.8	\[ x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{z \cdot z}}{y}, -1 \cdot y\right) \]
      associate-/l* [=>] 99.7	\[ x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{z}{\frac{y}{z}}}, -1 \cdot y\right) \]
      mul-1-neg [=>] 99.7	\[ x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, \color{blue}{-y}\right) \]

   4. Taylor expanded in x around 0 95.8%

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

   5. Simplified 99.7%

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

      [Start] 95.8	\[ \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right) \cdot x \]
      unpow2 [=>] 95.8	\[ \left(0.5 \cdot \frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot x \]
      associate-/l* [=>] 99.7	\[ \left(0.5 \cdot \color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot x \]

   if -1.99999999999999994e-304 < y

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf 94.3%

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

   3. Simplified 99.6%

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

      [Start] 94.3	\[ -0.5 \cdot \frac{{z}^{2} \cdot x}{y} + y \cdot x \]
      +-commutative [=>] 94.3	\[ \color{blue}{y \cdot x + -0.5 \cdot \frac{{z}^{2} \cdot x}{y}} \]
      fma-def [=>] 94.3	\[ \color{blue}{\mathsf{fma}\left(y, x, -0.5 \cdot \frac{{z}^{2} \cdot x}{y}\right)} \]
      associate-/l* [=>] 89.8	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \color{blue}{\frac{{z}^{2}}{\frac{y}{x}}}\right) \]
      unpow2 [=>] 89.8	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \frac{\color{blue}{z \cdot z}}{\frac{y}{x}}\right) \]
      associate-/r/ [=>] 94.7	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \color{blue}{\left(\frac{z \cdot z}{y} \cdot x\right)}\right) \]
      associate-/l* [=>] 99.6	\[ \mathsf{fma}\left(y, x, -0.5 \cdot \left(\color{blue}{\frac{z}{\frac{y}{z}}} \cdot x\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot \left(x \cdot -0.5\right) + y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-257}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Accuracy	52.6%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023137 
(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)))))