Average Error: 9.4 → 0.2
Time: 5.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r870363 = x;
        double r870364 = y;
        double r870365 = r870363 / r870364;
        double r870366 = 2.0;
        double r870367 = z;
        double r870368 = r870367 * r870366;
        double r870369 = 1.0;
        double r870370 = t;
        double r870371 = r870369 - r870370;
        double r870372 = r870368 * r870371;
        double r870373 = r870366 + r870372;
        double r870374 = r870370 * r870367;
        double r870375 = r870373 / r870374;
        double r870376 = r870365 + r870375;
        return r870376;
}
↓
double f(double x, double y, double z, double t) {
        double r870377 = 1.0;
        double r870378 = z;
        double r870379 = r870377 / r870378;
        double r870380 = 1.0;
        double r870381 = r870379 + r870380;
        double r870382 = t;
        double r870383 = r870381 - r870382;
        double r870384 = 2.0;
        double r870385 = r870384 / r870382;
        double r870386 = x;
        double r870387 = y;
        double r870388 = r870386 / r870387;
        double r870389 = fma(r870383, r870385, r870388);
        return r870389;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.4
Target0.1
Herbie0.2
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +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))))