Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Average Error: 9.1 → 0.1

Time: 5.0s

Precision: 64

Math

\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

↓

\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (x / y)) + ((double) (((double) (2.0 + ((double) (((double) (z * 2.0)) * ((double) (1.0 - t)))))) / ((double) (t * z))))));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (x / y)) + ((double) fma(2.0, ((double) (((double) (1.0 / t)) / z)), ((double) (((double) (2.0 * ((double) (1.0 / t)))) - 2.0))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.1

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

2. Taylor expanded around 0 0.1

   \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t \cdot z} + 2 \cdot \frac{1}{t}\right) - 2\right)}\]

3. Simplified 0.1

   \[\leadsto \frac{x}{y} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)}\]
4. Using strategy rm

5. Applied associate-/r* 0.1

   \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \color{blue}{\frac{\frac{1}{t}}{z}}, 2 \cdot \frac{1}{t} - 2\right)\]

6. Final simplification 0.1

   \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))