Average Error: 12.7 → 0.3
Time: 5.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)\]
\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{2}{r \cdot r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r15903 = 3.0;
        double r15904 = 2.0;
        double r15905 = r;
        double r15906 = r15905 * r15905;
        double r15907 = r15904 / r15906;
        double r15908 = r15903 + r15907;
        double r15909 = 0.125;
        double r15910 = v;
        double r15911 = r15904 * r15910;
        double r15912 = r15903 - r15911;
        double r15913 = r15909 * r15912;
        double r15914 = w;
        double r15915 = r15914 * r15914;
        double r15916 = r15915 * r15905;
        double r15917 = r15916 * r15905;
        double r15918 = r15913 * r15917;
        double r15919 = 1.0;
        double r15920 = r15919 - r15910;
        double r15921 = r15918 / r15920;
        double r15922 = r15908 - r15921;
        double r15923 = 4.5;
        double r15924 = r15922 - r15923;
        return r15924;
}
↓
double f(double v, double w, double r) {
        double r15925 = 2.0;
        double r15926 = r;
        double r15927 = r15926 * r15926;
        double r15928 = r15925 / r15927;
        double r15929 = 0.125;
        double r15930 = 3.0;
        double r15931 = v;
        double r15932 = r15925 * r15931;
        double r15933 = r15930 - r15932;
        double r15934 = r15929 * r15933;
        double r15935 = 1.0;
        double r15936 = r15935 - r15931;
        double r15937 = r15934 / r15936;
        double r15938 = w;
        double r15939 = r15938 * r15926;
        double r15940 = fabs(r15939);
        double r15941 = r15937 * r15940;
        double r15942 = r15941 * r15940;
        double r15943 = 4.5;
        double r15944 = r15943 - r15930;
        double r15945 = r15942 + r15944;
        double r15946 = r15928 - r15945;
        return r15946;
}

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-sqrt8.6

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

  5. Simplified8.6

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

  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+0.3

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

  11. Applied associate-*r*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 +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))