Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.1 → 0.3

Time: 7.7s

Precision: binary64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 7.1

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
2. Using strategy rm

3. Applied div-sub 7.1

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

5. Applied associate-/l* 2.2

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

6. Taylor expanded around 0 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020148 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))