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

Average Accuracy: 55.9% → 99.8%

Time: 14.5s

Precision: binary64

Cost: 7168

Math

\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

↓

\[\mathsf{fma}\left(\frac{x + z}{y}, z - x, -y\right) \cdot -0.5 \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

↓

(FPCore (x y z) :precision binary64 (* (fma (/ (+ x z) y) (- z x) (- y)) -0.5))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

↓

double code(double x, double y, double z) {
	return fma(((x + z) / y), (z - x), -y) * -0.5;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

↓

function code(x, y, z)
	return Float64(fma(Float64(Float64(x + z) / y), Float64(z - x), Float64(-y)) * -0.5)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

↓

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

Target

Original	55.9%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 55.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

2. Simplified 99.8%

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

   [Start] 55.9	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 55.9	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 55.9	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 55.9	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 55.9	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 55.9	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 55.9	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 55.9	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 55.9	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	63.8%
Cost	1108

\[\begin{array}{l} t_0 := \frac{x}{\frac{y}{x}} \cdot 0.5\\ t_1 := -0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-240}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	63.9%
Cost	1108

\[\begin{array}{l} t_0 := \frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-146}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-240}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-136}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	63.9%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-143}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-241}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 10^{-136}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	89.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-48} \lor \neg \left(z \leq 2.8 \cdot 10^{-18}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{\frac{y}{x}}\right) \cdot 0.5\\ \end{array} \]

Alternative 5
Accuracy	89.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-48} \lor \neg \left(z \leq 5.3 \cdot 10^{-20}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{x}} - y\right)\\ \end{array} \]

Alternative 6
Accuracy	63.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-147}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Accuracy	63.7%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-145}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-113}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8
Accuracy	63.6%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-150}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9
Accuracy	78.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-246} \lor \neg \left(y \leq 5.8 \cdot 10^{-255}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 10
Accuracy	89.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\left(y + \frac{x}{\frac{y}{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 11
Accuracy	99.8%
Cost	832

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

Alternative 12
Accuracy	63.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-109}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 13
Accuracy	57.8%
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))