Average Error: 9.8 → 0.1
Time: 16.8s
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 r35686244 = x;
        double r35686245 = y;
        double r35686246 = r35686244 / r35686245;
        double r35686247 = 2.0;
        double r35686248 = z;
        double r35686249 = r35686248 * r35686247;
        double r35686250 = 1.0;
        double r35686251 = t;
        double r35686252 = r35686250 - r35686251;
        double r35686253 = r35686249 * r35686252;
        double r35686254 = r35686247 + r35686253;
        double r35686255 = r35686251 * r35686248;
        double r35686256 = r35686254 / r35686255;
        double r35686257 = r35686246 + r35686256;
        return r35686257;
}
↓
double f(double x, double y, double z, double t) {
        double r35686258 = 2.0;
        double r35686259 = t;
        double r35686260 = r35686258 / r35686259;
        double r35686261 = z;
        double r35686262 = 1.0;
        double r35686263 = 1.0;
        double r35686264 = fma(r35686261, r35686262, r35686263);
        double r35686265 = r35686264 / r35686261;
        double r35686266 = x;
        double r35686267 = y;
        double r35686268 = r35686266 / r35686267;
        double r35686269 = fma(r35686260, r35686265, r35686268);
        double r35686270 = r35686269 - r35686258;
        return r35686270;
}

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))))