Average Error: 13.2 → 0.5
Time: 5.2s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\sqrt{\left|w \cdot r\right|}} \cdot \left({\left(\sqrt{\left|w \cdot r\right|}\right)}^{\frac{3}{2}} \cdot \left|w \cdot r\right|\right), 4.5\right) - 3\right)\]
\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(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\sqrt{\left|w \cdot r\right|}} \cdot \left({\left(\sqrt{\left|w \cdot r\right|}\right)}^{\frac{3}{2}} \cdot \left|w \cdot r\right|\right), 4.5\right) - 3\right)
double f(double v, double w, double r) {
        double r14962 = 3.0;
        double r14963 = 2.0;
        double r14964 = r;
        double r14965 = r14964 * r14964;
        double r14966 = r14963 / r14965;
        double r14967 = r14962 + r14966;
        double r14968 = 0.125;
        double r14969 = v;
        double r14970 = r14963 * r14969;
        double r14971 = r14962 - r14970;
        double r14972 = r14968 * r14971;
        double r14973 = w;
        double r14974 = r14973 * r14973;
        double r14975 = r14974 * r14964;
        double r14976 = r14975 * r14964;
        double r14977 = r14972 * r14976;
        double r14978 = 1.0;
        double r14979 = r14978 - r14969;
        double r14980 = r14977 / r14979;
        double r14981 = r14967 - r14980;
        double r14982 = 4.5;
        double r14983 = r14981 - r14982;
        return r14983;
}
↓
double f(double v, double w, double r) {
        double r14984 = 2.0;
        double r14985 = r;
        double r14986 = r14984 / r14985;
        double r14987 = r14986 / r14985;
        double r14988 = 0.125;
        double r14989 = 3.0;
        double r14990 = v;
        double r14991 = r14984 * r14990;
        double r14992 = r14989 - r14991;
        double r14993 = r14988 * r14992;
        double r14994 = 1.0;
        double r14995 = r14994 - r14990;
        double r14996 = r14993 / r14995;
        double r14997 = w;
        double r14998 = r14997 * r14985;
        double r14999 = fabs(r14998);
        double r15000 = sqrt(r14999);
        double r15001 = sqrt(r15000);
        double r15002 = 1.5;
        double r15003 = pow(r15000, r15002);
        double r15004 = r15003 * r14999;
        double r15005 = r15001 * r15004;
        double r15006 = 4.5;
        double r15007 = fma(r14996, r15005, r15006);
        double r15008 = r15007 - r14989;
        double r15009 = r14987 - r15008;
        return r15009;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

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. Simplified8.9

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right) - 3\right)}\]
  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. Simplified8.9

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

  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 add-sqr-sqrt0.4

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

  11. Applied associate-*l*0.4

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

  12. Simplified0.5

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

  14. Applied add-sqr-sqrt0.5

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

  15. Applied sqrt-prod0.6

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

  16. Applied associate-*l*0.6

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

  17. Simplified0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2020020 +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))