Average Error: 12.3 → 0.3
Time: 21.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\sqrt{{\left(\left|w \cdot r\right|\right)}^{2}} \cdot \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|\left|w \cdot r\right|\right|\right) + 4.5\right)\]
\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) - \left(\sqrt{{\left(\left|w \cdot r\right|\right)}^{2}} \cdot \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|\left|w \cdot r\right|\right|\right) + 4.5\right)
double f(double v, double w, double r) {
        double r25949 = 3.0;
        double r25950 = 2.0;
        double r25951 = r;
        double r25952 = r25951 * r25951;
        double r25953 = r25950 / r25952;
        double r25954 = r25949 + r25953;
        double r25955 = 0.125;
        double r25956 = v;
        double r25957 = r25950 * r25956;
        double r25958 = r25949 - r25957;
        double r25959 = r25955 * r25958;
        double r25960 = w;
        double r25961 = r25960 * r25960;
        double r25962 = r25961 * r25951;
        double r25963 = r25962 * r25951;
        double r25964 = r25959 * r25963;
        double r25965 = 1.0;
        double r25966 = r25965 - r25956;
        double r25967 = r25964 / r25966;
        double r25968 = r25954 - r25967;
        double r25969 = 4.5;
        double r25970 = r25968 - r25969;
        return r25970;
}
↓
double f(double v, double w, double r) {
        double r25971 = 3.0;
        double r25972 = 2.0;
        double r25973 = r;
        double r25974 = r25972 / r25973;
        double r25975 = r25974 / r25973;
        double r25976 = r25971 + r25975;
        double r25977 = w;
        double r25978 = r25977 * r25973;
        double r25979 = fabs(r25978);
        double r25980 = 2.0;
        double r25981 = pow(r25979, r25980);
        double r25982 = sqrt(r25981);
        double r25983 = 0.125;
        double r25984 = v;
        double r25985 = r25972 * r25984;
        double r25986 = r25971 - r25985;
        double r25987 = r25983 * r25986;
        double r25988 = 1.0;
        double r25989 = r25988 - r25984;
        double r25990 = r25987 / r25989;
        double r25991 = fabs(r25979);
        double r25992 = r25990 * r25991;
        double r25993 = r25982 * r25992;
        double r25994 = 4.5;
        double r25995 = r25993 + r25994;
        double r25996 = r25976 - r25995;
        return r25996;
}

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 12.3

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

  2. Simplified8.2

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

  4. Applied add-sqr-sqrt8.3

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

  5. Simplified8.2

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

  6. Simplified0.3

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

  8. Applied associate-/r*0.3

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

  10. Applied fma-udef0.3

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

  11. Simplified0.3

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

  13. Applied add-sqr-sqrt0.3

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

  14. Applied associate-*l*0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019306 +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))