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

Average Error: 9.2 → 0.1

Time: 4.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r711476 = x;
        double r711477 = y;
        double r711478 = r711476 / r711477;
        double r711479 = 2.0;
        double r711480 = z;
        double r711481 = r711480 * r711479;
        double r711482 = 1.0;
        double r711483 = t;
        double r711484 = r711482 - r711483;
        double r711485 = r711481 * r711484;
        double r711486 = r711479 + r711485;
        double r711487 = r711483 * r711480;
        double r711488 = r711486 / r711487;
        double r711489 = r711478 + r711488;
        return r711489;
}

↓

double f(double x, double y, double z, double t) {
        double r711490 = x;
        double r711491 = y;
        double r711492 = r711490 / r711491;
        double r711493 = 2.0;
        double r711494 = 1.0;
        double r711495 = t;
        double r711496 = z;
        double r711497 = r711495 * r711496;
        double r711498 = r711494 / r711497;
        double r711499 = r711494 / r711495;
        double r711500 = r711493 * r711499;
        double r711501 = r711500 - r711493;
        double r711502 = fma(r711493, r711498, r711501);
        double r711503 = r711492 + r711502;
        return r711503;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.2

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

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. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020001 +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))))