Average Error: 9.5 → 0.1
Time: 4.7s
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 r811537 = x;
        double r811538 = y;
        double r811539 = r811537 / r811538;
        double r811540 = 2.0;
        double r811541 = z;
        double r811542 = r811541 * r811540;
        double r811543 = 1.0;
        double r811544 = t;
        double r811545 = r811543 - r811544;
        double r811546 = r811542 * r811545;
        double r811547 = r811540 + r811546;
        double r811548 = r811544 * r811541;
        double r811549 = r811547 / r811548;
        double r811550 = r811539 + r811549;
        return r811550;
}
↓
double f(double x, double y, double z, double t) {
        double r811551 = 1.0;
        double r811552 = z;
        double r811553 = r811551 / r811552;
        double r811554 = 1.0;
        double r811555 = r811553 + r811554;
        double r811556 = t;
        double r811557 = r811555 - r811556;
        double r811558 = 2.0;
        double r811559 = r811558 / r811556;
        double r811560 = x;
        double r811561 = y;
        double r811562 = r811560 / r811561;
        double r811563 = fma(r811557, r811559, r811562);
        return r811563;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.5
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.5

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

  2. Simplified9.5

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020020 +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))))