Average Error: 12.7 → 0.4
Time: 7.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|w \cdot r\right|\right) \cdot \left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]
\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(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|w \cdot r\right|\right) \cdot \left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r18770 = 3.0;
        double r18771 = 2.0;
        double r18772 = r;
        double r18773 = r18772 * r18772;
        double r18774 = r18771 / r18773;
        double r18775 = r18770 + r18774;
        double r18776 = 0.125;
        double r18777 = v;
        double r18778 = r18771 * r18777;
        double r18779 = r18770 - r18778;
        double r18780 = r18776 * r18779;
        double r18781 = w;
        double r18782 = r18781 * r18781;
        double r18783 = r18782 * r18772;
        double r18784 = r18783 * r18772;
        double r18785 = r18780 * r18784;
        double r18786 = 1.0;
        double r18787 = r18786 - r18777;
        double r18788 = r18785 / r18787;
        double r18789 = r18775 - r18788;
        double r18790 = 4.5;
        double r18791 = r18789 - r18790;
        return r18791;
}
↓
double f(double v, double w, double r) {
        double r18792 = 2.0;
        double r18793 = r;
        double r18794 = r18792 / r18793;
        double r18795 = r18794 / r18793;
        double r18796 = 0.125;
        double r18797 = 3.0;
        double r18798 = v;
        double r18799 = r18792 * r18798;
        double r18800 = r18797 - r18799;
        double r18801 = r18796 * r18800;
        double r18802 = 1.0;
        double r18803 = r18802 - r18798;
        double r18804 = r18801 / r18803;
        double r18805 = sqrt(r18804);
        double r18806 = w;
        double r18807 = r18806 * r18793;
        double r18808 = fabs(r18807);
        double r18809 = r18805 * r18808;
        double r18810 = r18809 * r18809;
        double r18811 = 4.5;
        double r18812 = r18811 - r18797;
        double r18813 = r18810 + r18812;
        double r18814 = r18795 - r18813;
        return r18814;
}

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. Simplified8.6

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  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+0.3

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

  13. Applied add-sqr-sqrt0.4

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

  14. Applied unswap-sqr0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +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))