Average Error: 8.8 → 0.1
Time: 22.6s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(2, 1, \frac{2}{z}\right), \frac{1}{t}, \frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\mathsf{fma}\left(2, 1, \frac{2}{z}\right), \frac{1}{t}, \frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r451895 = x;
        double r451896 = y;
        double r451897 = r451895 / r451896;
        double r451898 = 2.0;
        double r451899 = z;
        double r451900 = r451899 * r451898;
        double r451901 = 1.0;
        double r451902 = t;
        double r451903 = r451901 - r451902;
        double r451904 = r451900 * r451903;
        double r451905 = r451898 + r451904;
        double r451906 = r451902 * r451899;
        double r451907 = r451905 / r451906;
        double r451908 = r451897 + r451907;
        return r451908;
}
↓
double f(double x, double y, double z, double t) {
        double r451909 = 2.0;
        double r451910 = 1.0;
        double r451911 = z;
        double r451912 = r451909 / r451911;
        double r451913 = fma(r451909, r451910, r451912);
        double r451914 = 1.0;
        double r451915 = t;
        double r451916 = r451914 / r451915;
        double r451917 = x;
        double r451918 = y;
        double r451919 = r451917 / r451918;
        double r451920 = r451919 - r451909;
        double r451921 = fma(r451913, r451916, r451920);
        return r451921;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original8.8
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 8.8

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

  2. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Applied fma-def0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 +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))))