Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Average Error: 5.5 → 0.1

Time: 5.9s

Precision: binary64

Cost: 704

Math

\[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}\]

↓

\[\frac{1 - x}{\frac{y}{\frac{3 - x}{3}}}\]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

↓

(FPCore (x y) :precision binary64 (/ (- 1.0 x) (/ y (/ (- 3.0 x) 3.0))))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

↓

double code(double x, double y) {
	return (1.0 - x) / (y / ((3.0 - x) / 3.0));
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.5
Target	0.1
Herbie	0.1

\[\frac{1 - x}{y} \cdot \frac{3 - x}{3}\]

Alternatives

Alternative 1
Error	0.2
Cost	704

\[\frac{\left(3 - x\right) \cdot \frac{1 - x}{y}}{3}\]

Alternative 2
Error	1.5
Cost	1218

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6757515790102016:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{3}\\ \mathbf{elif}\;x \leq 0.7992915933367786:\\ \;\;\;\;\frac{1}{y} - \frac{x}{y} \cdot 1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{-3 \cdot \frac{y}{x}}\\ \end{array}\]

Alternative 3
Error	2.0
Cost	1218

\[\begin{array}{l} \mathbf{if}\;x \leq -1.721800375688621:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{3}\\ \mathbf{elif}\;x \leq 4.920863345884939:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{-3 \cdot \frac{y}{x}}\\ \end{array}\]

Alternative 4
Error	2.1
Cost	1090

\[\begin{array}{l} \mathbf{if}\;x \leq -1.721800375688621:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{3}\\ \mathbf{elif}\;x \leq 0.5746400256711233:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \end{array}\]

Alternative 5
Error	2.1
Cost	776

\[\begin{array}{l} \mathbf{if}\;x \leq -1.721800375688621 \lor \neg \left(x \leq 0.5906865662186701\right):\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array}\]

Alternative 6
Error	20.8
Cost	192

\[\frac{1}{y}\]

Alternative 7
Error	60.6
Cost	385

\[\begin{array}{l} \mathbf{if}\;y \leq 4.948546683679213 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 8
Error	61.8
Cost	64

\[1\]

Derivation

1. Initial program 5.5

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}\]
2. Using strategy rm

3. Applied associate-/l*_binary64_12981 0.3

   \[\leadsto \color{blue}{\frac{1 - x}{\frac{y \cdot 3}{3 - x}}}\]

4. Simplified 0.1

   \[\leadsto \frac{1 - x}{\color{blue}{\frac{y}{\frac{3 - x}{3}}}}\]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{1 - x}{\frac{y}{\frac{3 - x}{3}}}}\]

6. Final simplification 0.1

   \[\leadsto \frac{1 - x}{\frac{y}{\frac{3 - x}{3}}}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))