Average Error: 12.9 → 0.4
Time: 21.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|\left|w \cdot r\right|\right|\right) \cdot \sqrt{{\left(\left|w \cdot r\right|\right)}^{2}}\right) - 4.5\]
\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(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|\left|w \cdot r\right|\right|\right) \cdot \sqrt{{\left(\left|w \cdot r\right|\right)}^{2}}\right) - 4.5
double f(double v, double w, double r) {
        double r23402 = 3.0;
        double r23403 = 2.0;
        double r23404 = r;
        double r23405 = r23404 * r23404;
        double r23406 = r23403 / r23405;
        double r23407 = r23402 + r23406;
        double r23408 = 0.125;
        double r23409 = v;
        double r23410 = r23403 * r23409;
        double r23411 = r23402 - r23410;
        double r23412 = r23408 * r23411;
        double r23413 = w;
        double r23414 = r23413 * r23413;
        double r23415 = r23414 * r23404;
        double r23416 = r23415 * r23404;
        double r23417 = r23412 * r23416;
        double r23418 = 1.0;
        double r23419 = r23418 - r23409;
        double r23420 = r23417 / r23419;
        double r23421 = r23407 - r23420;
        double r23422 = 4.5;
        double r23423 = r23421 - r23422;
        return r23423;
}
↓
double f(double v, double w, double r) {
        double r23424 = 3.0;
        double r23425 = 2.0;
        double r23426 = r;
        double r23427 = r23425 / r23426;
        double r23428 = r23427 / r23426;
        double r23429 = r23424 + r23428;
        double r23430 = 0.125;
        double r23431 = v;
        double r23432 = r23425 * r23431;
        double r23433 = r23424 - r23432;
        double r23434 = r23430 * r23433;
        double r23435 = 1.0;
        double r23436 = r23435 - r23431;
        double r23437 = r23434 / r23436;
        double r23438 = w;
        double r23439 = r23438 * r23426;
        double r23440 = fabs(r23439);
        double r23441 = fabs(r23440);
        double r23442 = r23437 * r23441;
        double r23443 = 2.0;
        double r23444 = pow(r23440, r23443);
        double r23445 = sqrt(r23444);
        double r23446 = r23442 * r23445;
        double r23447 = r23429 - r23446;
        double r23448 = 4.5;
        double r23449 = r23447 - r23448;
        return r23449;
}

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 12.9

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

  2. Simplified8.8

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

  4. Applied add-sqr-sqrt8.9

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

  5. Simplified8.8

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

  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 fma-udef0.4

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

  9. Applied associate--r+0.4

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

  10. Simplified0.4

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied associate-*r*0.4

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

  14. Simplified0.4

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

  16. Applied associate-/r*0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 +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))