Average Error: 12.8 → 0.4
Time: 24.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{1}{\frac{r}{\frac{2}{r}}} + 3\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\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(\frac{1}{\frac{r}{\frac{2}{r}}} + 3\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 r28301 = 3.0;
        double r28302 = 2.0;
        double r28303 = r;
        double r28304 = r28303 * r28303;
        double r28305 = r28302 / r28304;
        double r28306 = r28301 + r28305;
        double r28307 = 0.125;
        double r28308 = v;
        double r28309 = r28302 * r28308;
        double r28310 = r28301 - r28309;
        double r28311 = r28307 * r28310;
        double r28312 = w;
        double r28313 = r28312 * r28312;
        double r28314 = r28313 * r28303;
        double r28315 = r28314 * r28303;
        double r28316 = r28311 * r28315;
        double r28317 = 1.0;
        double r28318 = r28317 - r28308;
        double r28319 = r28316 / r28318;
        double r28320 = r28306 - r28319;
        double r28321 = 4.5;
        double r28322 = r28320 - r28321;
        return r28322;
}
↓
double f(double v, double w, double r) {
        double r28323 = 1.0;
        double r28324 = r;
        double r28325 = 2.0;
        double r28326 = r28325 / r28324;
        double r28327 = r28324 / r28326;
        double r28328 = r28323 / r28327;
        double r28329 = 3.0;
        double r28330 = r28328 + r28329;
        double r28331 = 0.125;
        double r28332 = v;
        double r28333 = r28325 * r28332;
        double r28334 = r28329 - r28333;
        double r28335 = r28331 * r28334;
        double r28336 = 1.0;
        double r28337 = r28336 - r28332;
        double r28338 = r28335 / r28337;
        double r28339 = w;
        double r28340 = r28339 * r28324;
        double r28341 = fabs(r28340);
        double r28342 = r28341 * r28341;
        double r28343 = 4.5;
        double r28344 = fma(r28338, r28342, r28343);
        double r28345 = r28330 - r28344;
        return r28345;
}

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 *-un-lft-identity0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied times-frac0.4

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

  13. Applied associate-/l*0.4

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

  14. Final simplification0.4

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

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))