Average Error: 12.9 → 0.6
Time: 11.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\left|w \cdot r\right|} \cdot {\left(\sqrt{\left|w \cdot r\right|}\right)}^{3}, 4.5\right)\]
\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{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\left|w \cdot r\right|} \cdot {\left(\sqrt{\left|w \cdot r\right|}\right)}^{3}, 4.5\right)
double f(double v, double w, double r) {
        double r15546 = 3.0;
        double r15547 = 2.0;
        double r15548 = r;
        double r15549 = r15548 * r15548;
        double r15550 = r15547 / r15549;
        double r15551 = r15546 + r15550;
        double r15552 = 0.125;
        double r15553 = v;
        double r15554 = r15547 * r15553;
        double r15555 = r15546 - r15554;
        double r15556 = r15552 * r15555;
        double r15557 = w;
        double r15558 = r15557 * r15557;
        double r15559 = r15558 * r15548;
        double r15560 = r15559 * r15548;
        double r15561 = r15556 * r15560;
        double r15562 = 1.0;
        double r15563 = r15562 - r15553;
        double r15564 = r15561 / r15563;
        double r15565 = r15551 - r15564;
        double r15566 = 4.5;
        double r15567 = r15565 - r15566;
        return r15567;
}
↓
double f(double v, double w, double r) {
        double r15568 = 3.0;
        double r15569 = 2.0;
        double r15570 = r;
        double r15571 = r15569 / r15570;
        double r15572 = r15571 / r15570;
        double r15573 = r15568 + r15572;
        double r15574 = 0.125;
        double r15575 = v;
        double r15576 = r15569 * r15575;
        double r15577 = r15568 - r15576;
        double r15578 = r15574 * r15577;
        double r15579 = 1.0;
        double r15580 = r15579 - r15575;
        double r15581 = r15578 / r15580;
        double r15582 = w;
        double r15583 = r15582 * r15570;
        double r15584 = fabs(r15583);
        double r15585 = sqrt(r15584);
        double r15586 = 3.0;
        double r15587 = pow(r15585, r15586);
        double r15588 = r15585 * r15587;
        double r15589 = 4.5;
        double r15590 = fma(r15581, r15588, r15589);
        double r15591 = r15573 - r15590;
        return r15591;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

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. Simplified8.5

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

  4. Applied add-sqr-sqrt8.5

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

  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 add-sqr-sqrt0.5

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

  11. Applied associate-*l*0.5

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019350 +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))