Average Error: 9.5 → 0.1
Time: 3.2s
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 r967324 = x;
        double r967325 = y;
        double r967326 = r967324 / r967325;
        double r967327 = 2.0;
        double r967328 = z;
        double r967329 = r967328 * r967327;
        double r967330 = 1.0;
        double r967331 = t;
        double r967332 = r967330 - r967331;
        double r967333 = r967329 * r967332;
        double r967334 = r967327 + r967333;
        double r967335 = r967331 * r967328;
        double r967336 = r967334 / r967335;
        double r967337 = r967326 + r967336;
        return r967337;
}
↓
double f(double x, double y, double z, double t) {
        double r967338 = x;
        double r967339 = y;
        double r967340 = r967338 / r967339;
        double r967341 = 2.0;
        double r967342 = 1.0;
        double r967343 = t;
        double r967344 = z;
        double r967345 = r967343 * r967344;
        double r967346 = r967342 / r967345;
        double r967347 = r967342 / r967343;
        double r967348 = r967341 * r967347;
        double r967349 = r967348 - r967341;
        double r967350 = fma(r967341, r967346, r967349);
        double r967351 = r967340 + r967350;
        return r967351;
}

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 2020056 +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))))