Average Error: 9.2 → 0.1
Time: 11.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(-2, 1, 2\right) + \left(\frac{2}{t \cdot z} + \left(\frac{2}{t} - 2\right)\right)\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \left(\mathsf{fma}\left(-2, 1, 2\right) + \left(\frac{2}{t \cdot z} + \left(\frac{2}{t} - 2\right)\right)\right)
double f(double x, double y, double z, double t) {
        double r843343 = x;
        double r843344 = y;
        double r843345 = r843343 / r843344;
        double r843346 = 2.0;
        double r843347 = z;
        double r843348 = r843347 * r843346;
        double r843349 = 1.0;
        double r843350 = t;
        double r843351 = r843349 - r843350;
        double r843352 = r843348 * r843351;
        double r843353 = r843346 + r843352;
        double r843354 = r843350 * r843347;
        double r843355 = r843353 / r843354;
        double r843356 = r843345 + r843355;
        return r843356;
}
↓
double f(double x, double y, double z, double t) {
        double r843357 = x;
        double r843358 = y;
        double r843359 = r843357 / r843358;
        double r843360 = 2.0;
        double r843361 = -r843360;
        double r843362 = 1.0;
        double r843363 = fma(r843361, r843362, r843360);
        double r843364 = t;
        double r843365 = z;
        double r843366 = r843364 * r843365;
        double r843367 = r843360 / r843366;
        double r843368 = r843360 / r843364;
        double r843369 = r843368 - r843360;
        double r843370 = r843367 + r843369;
        double r843371 = r843363 + r843370;
        double r843372 = r843359 + r843371;
        return r843372;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

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. Simplified0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\frac{2}{t \cdot z} + \left(\frac{2}{t} - 2\right)\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.1

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

  6. Applied add-sqr-sqrt32.1

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

  7. Applied prod-diff32.1

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

  8. Applied associate-+r+32.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +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))))