Average Error: 9.8 → 0.1
Time: 18.0s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(-1 + \frac{\mathsf{fma}\left(z, 1, 1\right)}{t \cdot z}, 2, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(-1 + \frac{\mathsf{fma}\left(z, 1, 1\right)}{t \cdot z}, 2, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r34564335 = x;
        double r34564336 = y;
        double r34564337 = r34564335 / r34564336;
        double r34564338 = 2.0;
        double r34564339 = z;
        double r34564340 = r34564339 * r34564338;
        double r34564341 = 1.0;
        double r34564342 = t;
        double r34564343 = r34564341 - r34564342;
        double r34564344 = r34564340 * r34564343;
        double r34564345 = r34564338 + r34564344;
        double r34564346 = r34564342 * r34564339;
        double r34564347 = r34564345 / r34564346;
        double r34564348 = r34564337 + r34564347;
        return r34564348;
}
↓
double f(double x, double y, double z, double t) {
        double r34564349 = -1.0;
        double r34564350 = z;
        double r34564351 = 1.0;
        double r34564352 = 1.0;
        double r34564353 = fma(r34564350, r34564351, r34564352);
        double r34564354 = t;
        double r34564355 = r34564354 * r34564350;
        double r34564356 = r34564353 / r34564355;
        double r34564357 = r34564349 + r34564356;
        double r34564358 = 2.0;
        double r34564359 = x;
        double r34564360 = y;
        double r34564361 = r34564359 / r34564360;
        double r34564362 = fma(r34564357, r34564358, r34564361);
        return r34564362;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.8
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.8

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

  2. Simplified0.1

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

  4. Applied sub-neg0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))