Average Error: 12.7 → 0.4
Time: 21.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(\frac{2}{{r}^{2}} - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}\right) + 3\right) - 4.5\]
\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(\frac{2}{{r}^{2}} - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}\right) + 3\right) - 4.5
double f(double v, double w, double r) {
        double r28685 = 3.0;
        double r28686 = 2.0;
        double r28687 = r;
        double r28688 = r28687 * r28687;
        double r28689 = r28686 / r28688;
        double r28690 = r28685 + r28689;
        double r28691 = 0.125;
        double r28692 = v;
        double r28693 = r28686 * r28692;
        double r28694 = r28685 - r28693;
        double r28695 = r28691 * r28694;
        double r28696 = w;
        double r28697 = r28696 * r28696;
        double r28698 = r28697 * r28687;
        double r28699 = r28698 * r28687;
        double r28700 = r28695 * r28699;
        double r28701 = 1.0;
        double r28702 = r28701 - r28692;
        double r28703 = r28700 / r28702;
        double r28704 = r28690 - r28703;
        double r28705 = 4.5;
        double r28706 = r28704 - r28705;
        return r28706;
}
↓
double f(double v, double w, double r) {
        double r28707 = 2.0;
        double r28708 = r;
        double r28709 = 2.0;
        double r28710 = pow(r28708, r28709);
        double r28711 = r28707 / r28710;
        double r28712 = 0.125;
        double r28713 = 3.0;
        double r28714 = v;
        double r28715 = r28707 * r28714;
        double r28716 = r28713 - r28715;
        double r28717 = r28712 * r28716;
        double r28718 = 1.0;
        double r28719 = r28718 - r28714;
        double r28720 = r28717 / r28719;
        double r28721 = w;
        double r28722 = r28721 * r28708;
        double r28723 = fabs(r28722);
        double r28724 = pow(r28723, r28709);
        double r28725 = r28720 * r28724;
        double r28726 = r28711 - r28725;
        double r28727 = r28726 + r28713;
        double r28728 = 4.5;
        double r28729 = r28727 - r28728;
        return r28729;
}

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 12.7

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

  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

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied fma-udef0.4

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

  11. Applied associate--r+0.4

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

  12. Simplified0.4

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

  13. Taylor expanded around 0 0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019305 +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))