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

Average Error: 9.3 → 0.1

Time: 22.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r653542 = x;
        double r653543 = y;
        double r653544 = r653542 / r653543;
        double r653545 = 2.0;
        double r653546 = z;
        double r653547 = r653546 * r653545;
        double r653548 = 1.0;
        double r653549 = t;
        double r653550 = r653548 - r653549;
        double r653551 = r653547 * r653550;
        double r653552 = r653545 + r653551;
        double r653553 = r653549 * r653546;
        double r653554 = r653552 / r653553;
        double r653555 = r653544 + r653554;
        return r653555;
}

↓

double f(double x, double y, double z, double t) {
        double r653556 = 2.0;
        double r653557 = 1.0;
        double r653558 = z;
        double r653559 = r653556 / r653558;
        double r653560 = fma(r653556, r653557, r653559);
        double r653561 = t;
        double r653562 = r653560 / r653561;
        double r653563 = x;
        double r653564 = y;
        double r653565 = r653563 / r653564;
        double r653566 = r653565 - r653556;
        double r653567 = r653562 + r653566;
        return r653567;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.3

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

2. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019306 +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))))