Average Error: 13.2 → 0.3
Time: 5.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)\]
\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
↓
\frac{\frac{2}{r}}{r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r17720 = 3.0;
        double r17721 = 2.0;
        double r17722 = r;
        double r17723 = r17722 * r17722;
        double r17724 = r17721 / r17723;
        double r17725 = r17720 + r17724;
        double r17726 = 0.125;
        double r17727 = v;
        double r17728 = r17721 * r17727;
        double r17729 = r17720 - r17728;
        double r17730 = r17726 * r17729;
        double r17731 = w;
        double r17732 = r17731 * r17731;
        double r17733 = r17732 * r17722;
        double r17734 = r17733 * r17722;
        double r17735 = r17730 * r17734;
        double r17736 = 1.0;
        double r17737 = r17736 - r17727;
        double r17738 = r17735 / r17737;
        double r17739 = r17725 - r17738;
        double r17740 = 4.5;
        double r17741 = r17739 - r17740;
        return r17741;
}
↓
double f(double v, double w, double r) {
        double r17742 = 2.0;
        double r17743 = r;
        double r17744 = r17742 / r17743;
        double r17745 = r17744 / r17743;
        double r17746 = 0.125;
        double r17747 = 3.0;
        double r17748 = v;
        double r17749 = r17742 * r17748;
        double r17750 = r17747 - r17749;
        double r17751 = r17746 * r17750;
        double r17752 = 1.0;
        double r17753 = r17752 - r17748;
        double r17754 = r17751 / r17753;
        double r17755 = w;
        double r17756 = r17755 * r17743;
        double r17757 = fabs(r17756);
        double r17758 = r17754 * r17757;
        double r17759 = r17758 * r17757;
        double r17760 = 4.5;
        double r17761 = r17760 - r17747;
        double r17762 = r17759 + r17761;
        double r17763 = r17745 - r17762;
        return r17763;
}

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 13.2

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

  2. Simplified8.8

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

  4. Applied add-sqr-sqrt8.9

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

  5. Simplified8.8

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied fma-udef0.4

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

  11. Applied associate--l+0.4

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

  13. Applied associate-*r*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019362 +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))