Average Error: 9.5 → 0.1
Time: 3.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\]
\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)
double f(double x, double y, double z, double t) {
        double r787471 = x;
        double r787472 = y;
        double r787473 = r787471 / r787472;
        double r787474 = 2.0;
        double r787475 = z;
        double r787476 = r787475 * r787474;
        double r787477 = 1.0;
        double r787478 = t;
        double r787479 = r787477 - r787478;
        double r787480 = r787476 * r787479;
        double r787481 = r787474 + r787480;
        double r787482 = r787478 * r787475;
        double r787483 = r787481 / r787482;
        double r787484 = r787473 + r787483;
        return r787484;
}
↓
double f(double x, double y, double z, double t) {
        double r787485 = x;
        double r787486 = y;
        double r787487 = r787485 / r787486;
        double r787488 = 2.0;
        double r787489 = 1.0;
        double r787490 = t;
        double r787491 = z;
        double r787492 = r787490 * r787491;
        double r787493 = r787489 / r787492;
        double r787494 = r787489 / r787490;
        double r787495 = r787488 * r787494;
        double r787496 = r787495 - r787488;
        double r787497 = fma(r787488, r787493, r787496);
        double r787498 = r787487 + r787497;
        return r787498;
}

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. 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. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 +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))))