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

Average Error: 28.3 → 0.2

Time: 8.2s

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	28.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 28.3

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

2. Simplified 0.2

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

   [Start] 28.3	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 28.3	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 28.3	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 28.3	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 28.3	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 28.3	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 28.3	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 28.3	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 28.3	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	6.6
Cost	1357

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-120} \lor \neg \left(z \cdot z \leq 10^{-79}\right) \land z \cdot z \leq 10^{-33}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	832

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

Alternative 3
Error	23.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-135}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-136}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot -0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6
Error	13.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 7
Error	27.0
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023054 
(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)))