Average Error: 12.7 → 0.4
Time: 22.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \mathsf{fma}\left(v, -2, {\left(\sqrt[3]{3}\right)}^{3}\right) + \left(v \cdot \left(\left(-2\right) + 2\right)\right) \cdot 0.125}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\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 \mathsf{fma}\left(v, -2, {\left(\sqrt[3]{3}\right)}^{3}\right) + \left(v \cdot \left(\left(-2\right) + 2\right)\right) \cdot 0.125}{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 r26757 = 3.0;
        double r26758 = 2.0;
        double r26759 = r;
        double r26760 = r26759 * r26759;
        double r26761 = r26758 / r26760;
        double r26762 = r26757 + r26761;
        double r26763 = 0.125;
        double r26764 = v;
        double r26765 = r26758 * r26764;
        double r26766 = r26757 - r26765;
        double r26767 = r26763 * r26766;
        double r26768 = w;
        double r26769 = r26768 * r26768;
        double r26770 = r26769 * r26759;
        double r26771 = r26770 * r26759;
        double r26772 = r26767 * r26771;
        double r26773 = 1.0;
        double r26774 = r26773 - r26764;
        double r26775 = r26772 / r26774;
        double r26776 = r26762 - r26775;
        double r26777 = 4.5;
        double r26778 = r26776 - r26777;
        return r26778;
}
↓
double f(double v, double w, double r) {
        double r26779 = 3.0;
        double r26780 = 2.0;
        double r26781 = r;
        double r26782 = r26780 / r26781;
        double r26783 = r26782 / r26781;
        double r26784 = r26779 + r26783;
        double r26785 = 0.125;
        double r26786 = v;
        double r26787 = -r26780;
        double r26788 = cbrt(r26779);
        double r26789 = 3.0;
        double r26790 = pow(r26788, r26789);
        double r26791 = fma(r26786, r26787, r26790);
        double r26792 = r26785 * r26791;
        double r26793 = r26787 + r26780;
        double r26794 = r26786 * r26793;
        double r26795 = r26794 * r26785;
        double r26796 = r26792 + r26795;
        double r26797 = 1.0;
        double r26798 = r26797 - r26786;
        double r26799 = r26796 / r26798;
        double r26800 = w;
        double r26801 = r26800 * r26781;
        double r26802 = fabs(r26801);
        double r26803 = r26802 * r26802;
        double r26804 = 4.5;
        double r26805 = fma(r26799, r26803, r26804);
        double r26806 = r26784 - r26805;
        return r26806;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

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

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

  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-cube-cbrt0.4

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

  11. Applied prod-diff0.4

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

  12. Applied distribute-lft-in0.4

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 +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))