Average Error: 12.6 → 0.4
Time: 5.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]
\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
↓
\frac{\frac{2}{r}}{r} - \left(\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r17383 = 3.0;
        double r17384 = 2.0;
        double r17385 = r;
        double r17386 = r17385 * r17385;
        double r17387 = r17384 / r17386;
        double r17388 = r17383 + r17387;
        double r17389 = 0.125;
        double r17390 = v;
        double r17391 = r17384 * r17390;
        double r17392 = r17383 - r17391;
        double r17393 = r17389 * r17392;
        double r17394 = w;
        double r17395 = r17394 * r17394;
        double r17396 = r17395 * r17385;
        double r17397 = r17396 * r17385;
        double r17398 = r17393 * r17397;
        double r17399 = 1.0;
        double r17400 = r17399 - r17390;
        double r17401 = r17398 / r17400;
        double r17402 = r17388 - r17401;
        double r17403 = 4.5;
        double r17404 = r17402 - r17403;
        return r17404;
}
↓
double f(double v, double w, double r) {
        double r17405 = 2.0;
        double r17406 = r;
        double r17407 = r17405 / r17406;
        double r17408 = r17407 / r17406;
        double r17409 = 0.125;
        double r17410 = 3.0;
        double r17411 = v;
        double r17412 = r17405 * r17411;
        double r17413 = r17410 - r17412;
        double r17414 = r17409 * r17413;
        double r17415 = 1.0;
        double r17416 = r17415 - r17411;
        double r17417 = r17414 / r17416;
        double r17418 = 3.0;
        double r17419 = pow(r17417, r17418);
        double r17420 = cbrt(r17419);
        double r17421 = w;
        double r17422 = r17421 * r17406;
        double r17423 = fabs(r17422);
        double r17424 = r17423 * r17423;
        double r17425 = r17420 * r17424;
        double r17426 = 4.5;
        double r17427 = r17426 - r17410;
        double r17428 = r17425 + r17427;
        double r17429 = r17408 - r17428;
        return r17429;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.6

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

  2. Simplified8.7

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied fma-udef0.4

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

  11. Applied associate--l+0.4

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

  13. Applied add-cbrt-cube7.3

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

  14. Applied add-cbrt-cube21.6

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

  15. Applied add-cbrt-cube21.6

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

  16. Applied cbrt-unprod21.6

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

  17. Applied cbrt-undiv21.6

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020024 +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))