Average Error: 9.8 → 0.1
Time: 16.7s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\frac{2}{t}, \frac{\mathsf{fma}\left(z, 1, 1\right)}{z}, \frac{x}{y}\right) - 2\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\frac{2}{t}, \frac{\mathsf{fma}\left(z, 1, 1\right)}{z}, \frac{x}{y}\right) - 2
double f(double x, double y, double z, double t) {
        double r34840336 = x;
        double r34840337 = y;
        double r34840338 = r34840336 / r34840337;
        double r34840339 = 2.0;
        double r34840340 = z;
        double r34840341 = r34840340 * r34840339;
        double r34840342 = 1.0;
        double r34840343 = t;
        double r34840344 = r34840342 - r34840343;
        double r34840345 = r34840341 * r34840344;
        double r34840346 = r34840339 + r34840345;
        double r34840347 = r34840343 * r34840340;
        double r34840348 = r34840346 / r34840347;
        double r34840349 = r34840338 + r34840348;
        return r34840349;
}
↓
double f(double x, double y, double z, double t) {
        double r34840350 = 2.0;
        double r34840351 = t;
        double r34840352 = r34840350 / r34840351;
        double r34840353 = z;
        double r34840354 = 1.0;
        double r34840355 = 1.0;
        double r34840356 = fma(r34840353, r34840354, r34840355);
        double r34840357 = r34840356 / r34840353;
        double r34840358 = x;
        double r34840359 = y;
        double r34840360 = r34840358 / r34840359;
        double r34840361 = fma(r34840352, r34840357, r34840360);
        double r34840362 = r34840361 - r34840350;
        return r34840362;
}

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

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

  3. Final simplification0.1

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

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))))