Average Error: 13.0 → 0.4
Time: 10.7s
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 r15697 = 3.0;
        double r15698 = 2.0;
        double r15699 = r;
        double r15700 = r15699 * r15699;
        double r15701 = r15698 / r15700;
        double r15702 = r15697 + r15701;
        double r15703 = 0.125;
        double r15704 = v;
        double r15705 = r15698 * r15704;
        double r15706 = r15697 - r15705;
        double r15707 = r15703 * r15706;
        double r15708 = w;
        double r15709 = r15708 * r15708;
        double r15710 = r15709 * r15699;
        double r15711 = r15710 * r15699;
        double r15712 = r15707 * r15711;
        double r15713 = 1.0;
        double r15714 = r15713 - r15704;
        double r15715 = r15712 / r15714;
        double r15716 = r15702 - r15715;
        double r15717 = 4.5;
        double r15718 = r15716 - r15717;
        return r15718;
}
↓
double f(double v, double w, double r) {
        double r15719 = 3.0;
        double r15720 = 2.0;
        double r15721 = r;
        double r15722 = r15720 / r15721;
        double r15723 = r15722 / r15721;
        double r15724 = r15719 + r15723;
        double r15725 = 0.125;
        double r15726 = v;
        double r15727 = r15720 * r15726;
        double r15728 = r15719 - r15727;
        double r15729 = r15725 * r15728;
        double r15730 = 1.0;
        double r15731 = r15730 - r15726;
        double r15732 = r15729 / r15731;
        double r15733 = w;
        double r15734 = r15733 * r15721;
        double r15735 = fabs(r15734);
        double r15736 = r15735 * r15735;
        double r15737 = 4.5;
        double r15738 = fma(r15732, r15736, r15737);
        double r15739 = r15724 - r15738;
        return r15739;
}

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. Simplified8.5

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

  4. Applied associate-*l*2.5

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

  6. Applied add-sqr-sqrt2.5

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

  7. Simplified2.5

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

  8. Simplified0.4

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

  10. Applied associate-/r*0.4

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

  11. 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 2020043 +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))