Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Average Error: 2.0 → 0.3

Time: 4.4s

Precision: 64

Math

\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]

↓

\[x - \left(\frac{\frac{y}{\left(t - z\right) + 1}}{\frac{1}{a}} - \frac{\frac{z}{\left(t - z\right) + 1}}{\frac{1}{a}}\right)\]

C

double code(double x, double y, double z, double t, double a) {
	return (x - ((y - z) / (((t - z) + 1.0) / a)));
}

↓

double code(double x, double y, double z, double t, double a) {
	return (x - (((y / ((t - z) + 1.0)) / (1.0 / a)) - ((z / ((t - z) + 1.0)) / (1.0 / a))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	2.0
Target	0.2
Herbie	0.3

\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

1. Initial program 2.0

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
2. Using strategy rm

3. Applied div-inv 2.0

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

4. Applied associate-/r* 0.3

   \[\leadsto x - \color{blue}{\frac{\frac{y - z}{\left(t - z\right) + 1}}{\frac{1}{a}}}\]
5. Using strategy rm

6. Applied div-sub 0.3

   \[\leadsto x - \frac{\color{blue}{\frac{y}{\left(t - z\right) + 1} - \frac{z}{\left(t - z\right) + 1}}}{\frac{1}{a}}\]

7. Applied div-sub 0.3

   \[\leadsto x - \color{blue}{\left(\frac{\frac{y}{\left(t - z\right) + 1}}{\frac{1}{a}} - \frac{\frac{z}{\left(t - z\right) + 1}}{\frac{1}{a}}\right)}\]

8. Final simplification 0.3

   \[\leadsto x - \left(\frac{\frac{y}{\left(t - z\right) + 1}}{\frac{1}{a}} - \frac{\frac{z}{\left(t - z\right) + 1}}{\frac{1}{a}}\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1) a))))