Average Error: 8.9 → 0.1
Time: 17.7s
Precision: 64
\[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{\mathsf{fma}\left(1.0, 2.0 \cdot z, 2.0\right)}{t \cdot z} + \frac{x}{y}\right) - 2.0\]
\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}
↓
\left(\frac{\mathsf{fma}\left(1.0, 2.0 \cdot z, 2.0\right)}{t \cdot z} + \frac{x}{y}\right) - 2.0
double f(double x, double y, double z, double t) {
        double r35525838 = x;
        double r35525839 = y;
        double r35525840 = r35525838 / r35525839;
        double r35525841 = 2.0;
        double r35525842 = z;
        double r35525843 = r35525842 * r35525841;
        double r35525844 = 1.0;
        double r35525845 = t;
        double r35525846 = r35525844 - r35525845;
        double r35525847 = r35525843 * r35525846;
        double r35525848 = r35525841 + r35525847;
        double r35525849 = r35525845 * r35525842;
        double r35525850 = r35525848 / r35525849;
        double r35525851 = r35525840 + r35525850;
        return r35525851;
}
↓
double f(double x, double y, double z, double t) {
        double r35525852 = 1.0;
        double r35525853 = 2.0;
        double r35525854 = z;
        double r35525855 = r35525853 * r35525854;
        double r35525856 = fma(r35525852, r35525855, r35525853);
        double r35525857 = t;
        double r35525858 = r35525857 * r35525854;
        double r35525859 = r35525856 / r35525858;
        double r35525860 = x;
        double r35525861 = y;
        double r35525862 = r35525860 / r35525861;
        double r35525863 = r35525859 + r35525862;
        double r35525864 = r35525863 - r35525853;
        return r35525864;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original8.9
Target0.1
Herbie0.1
\[\frac{\frac{2.0}{z} + 2.0}{t} - \left(2.0 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 8.9

    \[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2.0}{t}, \frac{\mathsf{fma}\left(z, 1.0, 1\right)}{z}, \frac{x}{y}\right) - 2.0}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{\left(\frac{2.0}{t} \cdot \frac{\mathsf{fma}\left(z, 1.0, 1\right)}{z} + \frac{x}{y}\right)} - 2.0\]

  5. Simplified0.1

    \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(1.0, z \cdot 2.0, 2.0\right)}{t \cdot z}} + \frac{x}{y}\right) - 2.0\]

  6. Final simplification0.1

    \[\leadsto \left(\frac{\mathsf{fma}\left(1.0, 2.0 \cdot z, 2.0\right)}{t \cdot z} + \frac{x}{y}\right) - 2.0\]

Reproduce

herbie shell --seed 2019164 +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))))