Average Error: 12.8 → 0.4
Time: 22.7s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{1}{\frac{r}{\frac{2}{r}}}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(3 + \frac{1}{\frac{r}{\frac{2}{r}}}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)
double f(double v, double w, double r) {
        double r27956 = 3.0;
        double r27957 = 2.0;
        double r27958 = r;
        double r27959 = r27958 * r27958;
        double r27960 = r27957 / r27959;
        double r27961 = r27956 + r27960;
        double r27962 = 0.125;
        double r27963 = v;
        double r27964 = r27957 * r27963;
        double r27965 = r27956 - r27964;
        double r27966 = r27962 * r27965;
        double r27967 = w;
        double r27968 = r27967 * r27967;
        double r27969 = r27968 * r27958;
        double r27970 = r27969 * r27958;
        double r27971 = r27966 * r27970;
        double r27972 = 1.0;
        double r27973 = r27972 - r27963;
        double r27974 = r27971 / r27973;
        double r27975 = r27961 - r27974;
        double r27976 = 4.5;
        double r27977 = r27975 - r27976;
        return r27977;
}
↓
double f(double v, double w, double r) {
        double r27978 = 3.0;
        double r27979 = 1.0;
        double r27980 = r;
        double r27981 = 2.0;
        double r27982 = r27981 / r27980;
        double r27983 = r27980 / r27982;
        double r27984 = r27979 / r27983;
        double r27985 = r27978 + r27984;
        double r27986 = 0.125;
        double r27987 = v;
        double r27988 = r27981 * r27987;
        double r27989 = r27978 - r27988;
        double r27990 = r27986 * r27989;
        double r27991 = 1.0;
        double r27992 = r27991 - r27987;
        double r27993 = r27990 / r27992;
        double r27994 = w;
        double r27995 = r27994 * r27980;
        double r27996 = fabs(r27995);
        double r27997 = r27996 * r27996;
        double r27998 = 4.5;
        double r27999 = fma(r27993, r27997, r27998);
        double r28000 = r27985 - r27999;
        return r28000;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.8

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

  2. Simplified8.5

    \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt8.6

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}, 4.5\right)\]

  5. Simplified8.5

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\left|w \cdot r\right|} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}, 4.5\right)\]

  6. Simplified0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \color{blue}{\left|w \cdot r\right|}, 4.5\right)\]
  7. Using strategy rm

  8. Applied associate-/r*0.4

    \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
  9. Using strategy rm

  10. Applied clear-num0.4

    \[\leadsto \left(3 + \color{blue}{\frac{1}{\frac{r}{\frac{2}{r}}}}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]

  11. Final simplification0.4

    \[\leadsto \left(3 + \frac{1}{\frac{r}{\frac{2}{r}}}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))