Average Error: 13.0 → 0.4
Time: 20.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\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}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)
double f(double v, double w, double r) {
        double r24586 = 3.0;
        double r24587 = 2.0;
        double r24588 = r;
        double r24589 = r24588 * r24588;
        double r24590 = r24587 / r24589;
        double r24591 = r24586 + r24590;
        double r24592 = 0.125;
        double r24593 = v;
        double r24594 = r24587 * r24593;
        double r24595 = r24586 - r24594;
        double r24596 = r24592 * r24595;
        double r24597 = w;
        double r24598 = r24597 * r24597;
        double r24599 = r24598 * r24588;
        double r24600 = r24599 * r24588;
        double r24601 = r24596 * r24600;
        double r24602 = 1.0;
        double r24603 = r24602 - r24593;
        double r24604 = r24601 / r24603;
        double r24605 = r24591 - r24604;
        double r24606 = 4.5;
        double r24607 = r24605 - r24606;
        return r24607;
}
↓
double f(double v, double w, double r) {
        double r24608 = 3.0;
        double r24609 = 2.0;
        double r24610 = r;
        double r24611 = r24609 / r24610;
        double r24612 = r24611 / r24610;
        double r24613 = r24608 + r24612;
        double r24614 = 0.125;
        double r24615 = v;
        double r24616 = r24609 * r24615;
        double r24617 = r24608 - r24616;
        double r24618 = r24614 * r24617;
        double r24619 = 1.0;
        double r24620 = r24619 - r24615;
        double r24621 = r24618 / r24620;
        double r24622 = w;
        double r24623 = r24622 * r24610;
        double r24624 = fabs(r24623);
        double r24625 = r24624 * r24624;
        double r24626 = 4.5;
        double r24627 = fma(r24621, r24625, r24626);
        double r24628 = r24613 - r24627;
        return r24628;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 13.0

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

  2. Simplified8.7

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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))