Average Error: 12.8 → 0.4
Time: 21.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{{\left(\left|w \cdot r\right|\right)}^{2} \cdot 0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}\right)\]
\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
↓
\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{{\left(\left|w \cdot r\right|\right)}^{2} \cdot 0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}\right)
double f(double v, double w, double r) {
        double r25334 = 3.0;
        double r25335 = 2.0;
        double r25336 = r;
        double r25337 = r25336 * r25336;
        double r25338 = r25335 / r25337;
        double r25339 = r25334 + r25338;
        double r25340 = 0.125;
        double r25341 = v;
        double r25342 = r25335 * r25341;
        double r25343 = r25334 - r25342;
        double r25344 = r25340 * r25343;
        double r25345 = w;
        double r25346 = r25345 * r25345;
        double r25347 = r25346 * r25336;
        double r25348 = r25347 * r25336;
        double r25349 = r25344 * r25348;
        double r25350 = 1.0;
        double r25351 = r25350 - r25341;
        double r25352 = r25349 / r25351;
        double r25353 = r25339 - r25352;
        double r25354 = 4.5;
        double r25355 = r25353 - r25354;
        return r25355;
}
↓
double f(double v, double w, double r) {
        double r25356 = 3.0;
        double r25357 = 2.0;
        double r25358 = r;
        double r25359 = r25358 * r25358;
        double r25360 = r25357 / r25359;
        double r25361 = r25356 + r25360;
        double r25362 = 4.5;
        double r25363 = w;
        double r25364 = r25363 * r25358;
        double r25365 = fabs(r25364);
        double r25366 = 2.0;
        double r25367 = pow(r25365, r25366);
        double r25368 = 0.125;
        double r25369 = r25367 * r25368;
        double r25370 = 1.0;
        double r25371 = v;
        double r25372 = r25370 - r25371;
        double r25373 = -r25357;
        double r25374 = fma(r25373, r25371, r25356);
        double r25375 = r25372 / r25374;
        double r25376 = r25369 / r25375;
        double r25377 = r25362 + r25376;
        double r25378 = r25361 - r25377;
        return r25378;
}

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. Simplified12.7

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

  4. Applied add-sqr-sqrt12.8

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

  5. Simplified12.7

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

  6. Simplified0.4

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

  8. Applied fma-udef0.4

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

  9. Simplified0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied associate-*r*0.4

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

  13. Simplified0.4

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

  15. Applied pow10.4

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

  16. Applied pow10.4

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

  17. Applied pow10.4

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

  18. Applied pow-prod-down0.4

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

  19. Applied pow-prod-down0.4

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

  20. Simplified0.4

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

  21. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))