Rosa's TurbineBenchmark

Average Error: 12.3 → 0.6

Time: 10.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double v, double w, double r) {
        double r15259 = 3.0;
        double r15260 = 2.0;
        double r15261 = r;
        double r15262 = r15261 * r15261;
        double r15263 = r15260 / r15262;
        double r15264 = r15259 + r15263;
        double r15265 = 0.125;
        double r15266 = v;
        double r15267 = r15260 * r15266;
        double r15268 = r15259 - r15267;
        double r15269 = r15265 * r15268;
        double r15270 = w;
        double r15271 = r15270 * r15270;
        double r15272 = r15271 * r15261;
        double r15273 = r15272 * r15261;
        double r15274 = r15269 * r15273;
        double r15275 = 1.0;
        double r15276 = r15275 - r15266;
        double r15277 = r15274 / r15276;
        double r15278 = r15264 - r15277;
        double r15279 = 4.5;
        double r15280 = r15278 - r15279;
        return r15280;
}

↓

double f(double v, double w, double r) {
        double r15281 = 3.0;
        double r15282 = 2.0;
        double r15283 = r;
        double r15284 = r15282 / r15283;
        double r15285 = r15284 / r15283;
        double r15286 = r15281 + r15285;
        double r15287 = 0.125;
        double r15288 = v;
        double r15289 = r15282 * r15288;
        double r15290 = r15281 - r15289;
        double r15291 = r15287 * r15290;
        double r15292 = 1.0;
        double r15293 = r15292 - r15288;
        double r15294 = r15291 / r15293;
        double r15295 = w;
        double r15296 = r15295 * r15283;
        double r15297 = fabs(r15296);
        double r15298 = 2.0;
        double r15299 = pow(r15297, r15298);
        double r15300 = cbrt(r15299);
        double r15301 = r15300 * r15300;
        double r15302 = r15301 * r15300;
        double r15303 = 4.5;
        double r15304 = fma(r15294, r15302, r15303);
        double r15305 = r15286 - r15304;
        return r15305;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

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. Simplified 8.2

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

4. Applied associate-*l* 2.5

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

6. Applied add-sqr-sqrt 2.6

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

7. Simplified 2.6

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

8. Simplified 0.3

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

10. Applied associate-/r* 0.3

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

12. Applied add-cube-cbrt 0.6

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

13. Simplified 0.6

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

14. Simplified 0.6

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

15. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020042 +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))