Average Error: 9.3 → 0.1
Time: 4.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r776838 = x;
        double r776839 = y;
        double r776840 = r776838 / r776839;
        double r776841 = 2.0;
        double r776842 = z;
        double r776843 = r776842 * r776841;
        double r776844 = 1.0;
        double r776845 = t;
        double r776846 = r776844 - r776845;
        double r776847 = r776843 * r776846;
        double r776848 = r776841 + r776847;
        double r776849 = r776845 * r776842;
        double r776850 = r776848 / r776849;
        double r776851 = r776840 + r776850;
        return r776851;
}
↓
double f(double x, double y, double z, double t) {
        double r776852 = 1.0;
        double r776853 = z;
        double r776854 = r776852 / r776853;
        double r776855 = 1.0;
        double r776856 = r776854 + r776855;
        double r776857 = t;
        double r776858 = r776856 - r776857;
        double r776859 = 2.0;
        double r776860 = r776859 / r776857;
        double r776861 = x;
        double r776862 = y;
        double r776863 = r776861 / r776862;
        double r776864 = fma(r776858, r776860, r776863);
        return r776864;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

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

  2. Simplified9.3

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +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))))