Average Error: 11.8 → 0.3
Time: 27.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 - \mathsf{fma}\left(\frac{\left(w \cdot r\right) \cdot 0.125}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right), w \cdot r, 4.5\right)\right) + \frac{\frac{2}{r}}{r}\]
\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 - \mathsf{fma}\left(\frac{\left(w \cdot r\right) \cdot 0.125}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right), w \cdot r, 4.5\right)\right) + \frac{\frac{2}{r}}{r}
double f(double v, double w, double r) {
        double r1123382 = 3.0;
        double r1123383 = 2.0;
        double r1123384 = r;
        double r1123385 = r1123384 * r1123384;
        double r1123386 = r1123383 / r1123385;
        double r1123387 = r1123382 + r1123386;
        double r1123388 = 0.125;
        double r1123389 = v;
        double r1123390 = r1123383 * r1123389;
        double r1123391 = r1123382 - r1123390;
        double r1123392 = r1123388 * r1123391;
        double r1123393 = w;
        double r1123394 = r1123393 * r1123393;
        double r1123395 = r1123394 * r1123384;
        double r1123396 = r1123395 * r1123384;
        double r1123397 = r1123392 * r1123396;
        double r1123398 = 1.0;
        double r1123399 = r1123398 - r1123389;
        double r1123400 = r1123397 / r1123399;
        double r1123401 = r1123387 - r1123400;
        double r1123402 = 4.5;
        double r1123403 = r1123401 - r1123402;
        return r1123403;
}
↓
double f(double v, double w, double r) {
        double r1123404 = 3.0;
        double r1123405 = w;
        double r1123406 = r;
        double r1123407 = r1123405 * r1123406;
        double r1123408 = 0.125;
        double r1123409 = r1123407 * r1123408;
        double r1123410 = 1.0;
        double r1123411 = v;
        double r1123412 = r1123410 - r1123411;
        double r1123413 = r1123409 / r1123412;
        double r1123414 = -2.0;
        double r1123415 = fma(r1123414, r1123411, r1123404);
        double r1123416 = r1123413 * r1123415;
        double r1123417 = 4.5;
        double r1123418 = fma(r1123416, r1123407, r1123417);
        double r1123419 = r1123404 - r1123418;
        double r1123420 = 2.0;
        double r1123421 = r1123420 / r1123406;
        double r1123422 = r1123421 / r1123406;
        double r1123423 = r1123419 + r1123422;
        return r1123423;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 11.8

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

  2. Simplified0.3

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

  4. Applied associate-/r/0.3

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

  6. Applied associate-/r*0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))