Average Error: 12.8 → 0.3
Time: 8.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[3 + \left(\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\right)\]
\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
↓
3 + \left(\frac{\frac{2}{r}}{r} - \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)\right)
double f(double v, double w, double r) {
        double r17152 = 3.0;
        double r17153 = 2.0;
        double r17154 = r;
        double r17155 = r17154 * r17154;
        double r17156 = r17153 / r17155;
        double r17157 = r17152 + r17156;
        double r17158 = 0.125;
        double r17159 = v;
        double r17160 = r17153 * r17159;
        double r17161 = r17152 - r17160;
        double r17162 = r17158 * r17161;
        double r17163 = w;
        double r17164 = r17163 * r17163;
        double r17165 = r17164 * r17154;
        double r17166 = r17165 * r17154;
        double r17167 = r17162 * r17166;
        double r17168 = 1.0;
        double r17169 = r17168 - r17159;
        double r17170 = r17167 / r17169;
        double r17171 = r17157 - r17170;
        double r17172 = 4.5;
        double r17173 = r17171 - r17172;
        return r17173;
}
↓
double f(double v, double w, double r) {
        double r17174 = 3.0;
        double r17175 = 2.0;
        double r17176 = r;
        double r17177 = r17175 / r17176;
        double r17178 = r17177 / r17176;
        double r17179 = 0.125;
        double r17180 = v;
        double r17181 = r17175 * r17180;
        double r17182 = r17174 - r17181;
        double r17183 = r17179 * r17182;
        double r17184 = 1.0;
        double r17185 = r17184 - r17180;
        double r17186 = r17183 / r17185;
        double r17187 = w;
        double r17188 = r17187 * r17176;
        double r17189 = fabs(r17188);
        double r17190 = r17189 * r17189;
        double r17191 = 4.5;
        double r17192 = fma(r17186, r17190, r17191);
        double r17193 = r17178 - r17192;
        double r17194 = r17174 + r17193;
        return r17194;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.8

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

  2. Simplified8.4

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

  4. Applied associate-*l*2.6

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

  5. Simplified2.6

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

  7. Applied add-sqr-sqrt2.6

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

  8. Simplified2.6

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

  9. Simplified0.3

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

  11. Applied associate-/r*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +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))