Average Error: 12.7 → 0.4
Time: 26.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} + \left(3 - \mathsf{fma}\left(\frac{\sqrt{0.125}}{1 - v} \cdot \left(\left(\sqrt{\sqrt{0.125}} \cdot \left(w \cdot r\right)\right) \cdot \left(\sqrt{\sqrt{0.125}} \cdot \left(w \cdot r\right)\right)\right), 3 - 2 \cdot v, 4.5\right)\right)\]
\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
↓
\frac{2}{r \cdot r} + \left(3 - \mathsf{fma}\left(\frac{\sqrt{0.125}}{1 - v} \cdot \left(\left(\sqrt{\sqrt{0.125}} \cdot \left(w \cdot r\right)\right) \cdot \left(\sqrt{\sqrt{0.125}} \cdot \left(w \cdot r\right)\right)\right), 3 - 2 \cdot v, 4.5\right)\right)
double f(double v, double w, double r) {
        double r1344535 = 3.0;
        double r1344536 = 2.0;
        double r1344537 = r;
        double r1344538 = r1344537 * r1344537;
        double r1344539 = r1344536 / r1344538;
        double r1344540 = r1344535 + r1344539;
        double r1344541 = 0.125;
        double r1344542 = v;
        double r1344543 = r1344536 * r1344542;
        double r1344544 = r1344535 - r1344543;
        double r1344545 = r1344541 * r1344544;
        double r1344546 = w;
        double r1344547 = r1344546 * r1344546;
        double r1344548 = r1344547 * r1344537;
        double r1344549 = r1344548 * r1344537;
        double r1344550 = r1344545 * r1344549;
        double r1344551 = 1.0;
        double r1344552 = r1344551 - r1344542;
        double r1344553 = r1344550 / r1344552;
        double r1344554 = r1344540 - r1344553;
        double r1344555 = 4.5;
        double r1344556 = r1344554 - r1344555;
        return r1344556;
}
↓
double f(double v, double w, double r) {
        double r1344557 = 2.0;
        double r1344558 = r;
        double r1344559 = r1344558 * r1344558;
        double r1344560 = r1344557 / r1344559;
        double r1344561 = 3.0;
        double r1344562 = 0.125;
        double r1344563 = sqrt(r1344562);
        double r1344564 = 1.0;
        double r1344565 = v;
        double r1344566 = r1344564 - r1344565;
        double r1344567 = r1344563 / r1344566;
        double r1344568 = sqrt(r1344563);
        double r1344569 = w;
        double r1344570 = r1344569 * r1344558;
        double r1344571 = r1344568 * r1344570;
        double r1344572 = r1344571 * r1344571;
        double r1344573 = r1344567 * r1344572;
        double r1344574 = r1344557 * r1344565;
        double r1344575 = r1344561 - r1344574;
        double r1344576 = 4.5;
        double r1344577 = fma(r1344573, r1344575, r1344576);
        double r1344578 = r1344561 - r1344577;
        double r1344579 = r1344560 + r1344578;
        return r1344579;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.7

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

  2. Simplified0.4

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

  4. Applied div-inv0.4

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

  5. Applied add-sqr-sqrt0.6

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied sqrt-prod0.4

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

  11. Applied unswap-sqr0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 +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))