Average Error: 12.5 → 0.5
Time: 7.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\mathsf{fma}\left(-\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{2}{r \cdot r}\right) - \left(4.5 - 3\right)\]
\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
↓
\mathsf{fma}\left(-\sqrt[3]{{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)}^{3}}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{2}{r \cdot r}\right) - \left(4.5 - 3\right)
double f(double v, double w, double r) {
        double r18924 = 3.0;
        double r18925 = 2.0;
        double r18926 = r;
        double r18927 = r18926 * r18926;
        double r18928 = r18925 / r18927;
        double r18929 = r18924 + r18928;
        double r18930 = 0.125;
        double r18931 = v;
        double r18932 = r18925 * r18931;
        double r18933 = r18924 - r18932;
        double r18934 = r18930 * r18933;
        double r18935 = w;
        double r18936 = r18935 * r18935;
        double r18937 = r18936 * r18926;
        double r18938 = r18937 * r18926;
        double r18939 = r18934 * r18938;
        double r18940 = 1.0;
        double r18941 = r18940 - r18931;
        double r18942 = r18939 / r18941;
        double r18943 = r18929 - r18942;
        double r18944 = 4.5;
        double r18945 = r18943 - r18944;
        return r18945;
}
↓
double f(double v, double w, double r) {
        double r18946 = 0.125;
        double r18947 = 3.0;
        double r18948 = 2.0;
        double r18949 = v;
        double r18950 = r18948 * r18949;
        double r18951 = r18947 - r18950;
        double r18952 = r18946 * r18951;
        double r18953 = 1.0;
        double r18954 = r18953 - r18949;
        double r18955 = r18952 / r18954;
        double r18956 = 3.0;
        double r18957 = pow(r18955, r18956);
        double r18958 = cbrt(r18957);
        double r18959 = -r18958;
        double r18960 = w;
        double r18961 = r;
        double r18962 = r18960 * r18961;
        double r18963 = fabs(r18962);
        double r18964 = 2.0;
        double r18965 = pow(r18963, r18964);
        double r18966 = r18961 * r18961;
        double r18967 = r18948 / r18966;
        double r18968 = fma(r18959, r18965, r18967);
        double r18969 = 4.5;
        double r18970 = r18969 - r18947;
        double r18971 = r18968 - r18970;
        return r18971;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.5

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

  2. Simplified8.1

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

  4. Applied add-sqr-sqrt8.1

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

  5. Simplified8.1

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

  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 fma-udef0.4

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

  9. Applied associate--l+0.4

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

  10. Applied associate--r+0.4

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

  11. Simplified0.4

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

  13. Applied add-cbrt-cube7.3

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

  14. Applied add-cbrt-cube21.6

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

  15. Applied add-cbrt-cube21.6

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

  16. Applied cbrt-unprod21.6

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

  17. Applied cbrt-undiv21.6

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

  18. Simplified0.5

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

  19. Final simplification0.5

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

Reproduce

herbie shell --seed 2020100 +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))