Average Error: 12.6 → 0.3
Time: 6.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)\]
\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{\frac{2}{r}}{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 r19017 = 3.0;
        double r19018 = 2.0;
        double r19019 = r;
        double r19020 = r19019 * r19019;
        double r19021 = r19018 / r19020;
        double r19022 = r19017 + r19021;
        double r19023 = 0.125;
        double r19024 = v;
        double r19025 = r19018 * r19024;
        double r19026 = r19017 - r19025;
        double r19027 = r19023 * r19026;
        double r19028 = w;
        double r19029 = r19028 * r19028;
        double r19030 = r19029 * r19019;
        double r19031 = r19030 * r19019;
        double r19032 = r19027 * r19031;
        double r19033 = 1.0;
        double r19034 = r19033 - r19024;
        double r19035 = r19032 / r19034;
        double r19036 = r19022 - r19035;
        double r19037 = 4.5;
        double r19038 = r19036 - r19037;
        return r19038;
}
↓
double f(double v, double w, double r) {
        double r19039 = 2.0;
        double r19040 = r;
        double r19041 = r19039 / r19040;
        double r19042 = r19041 / r19040;
        double r19043 = 0.125;
        double r19044 = 3.0;
        double r19045 = v;
        double r19046 = r19039 * r19045;
        double r19047 = r19044 - r19046;
        double r19048 = r19043 * r19047;
        double r19049 = 1.0;
        double r19050 = r19049 - r19045;
        double r19051 = r19048 / r19050;
        double r19052 = w;
        double r19053 = r19052 * r19040;
        double r19054 = fabs(r19053);
        double r19055 = r19051 * r19054;
        double r19056 = r19055 * r19054;
        double r19057 = 4.5;
        double r19058 = r19057 - r19044;
        double r19059 = r19056 + r19058;
        double r19060 = r19042 - r19059;
        return r19060;
}

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 12.6

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

  2. Simplified8.5

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right) - 3\right)}\]
  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 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 associate-*r*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019354 +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))