Average Error: 0.4 → 0.1
Time: 5.7m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{2}}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot t}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{2}}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot t}
double f(double v, double t) {
        double r49800620 = 1.0;
        double r49800621 = 5.0;
        double r49800622 = v;
        double r49800623 = r49800622 * r49800622;
        double r49800624 = r49800621 * r49800623;
        double r49800625 = r49800620 - r49800624;
        double r49800626 = atan2(1.0, 0.0);
        double r49800627 = t;
        double r49800628 = r49800626 * r49800627;
        double r49800629 = 2.0;
        double r49800630 = 3.0;
        double r49800631 = r49800630 * r49800623;
        double r49800632 = r49800620 - r49800631;
        double r49800633 = r49800629 * r49800632;
        double r49800634 = sqrt(r49800633);
        double r49800635 = r49800628 * r49800634;
        double r49800636 = r49800620 - r49800623;
        double r49800637 = r49800635 * r49800636;
        double r49800638 = r49800625 / r49800637;
        return r49800638;
}
↓
double f(double v, double t) {
        double r49800639 = v;
        double r49800640 = r49800639 * r49800639;
        double r49800641 = -5.0;
        double r49800642 = 1.0;
        double r49800643 = fma(r49800640, r49800641, r49800642);
        double r49800644 = r49800642 - r49800640;
        double r49800645 = r49800643 / r49800644;
        double r49800646 = atan2(1.0, 0.0);
        double r49800647 = 2.0;
        double r49800648 = sqrt(r49800647);
        double r49800649 = r49800646 * r49800648;
        double r49800650 = r49800645 / r49800649;
        double r49800651 = -3.0;
        double r49800652 = fma(r49800640, r49800651, r49800642);
        double r49800653 = sqrt(r49800652);
        double r49800654 = t;
        double r49800655 = r49800653 * r49800654;
        double r49800656 = r49800650 / r49800655;
        return r49800656;
}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{1 - v \cdot v}}{\sqrt{2 \cdot \mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)} \cdot \left(t \cdot \pi\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{1 - v \cdot v}}{\sqrt{2 \cdot \mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)}}}{t \cdot \pi}}\]
  5. Using strategy rm

  6. Applied sqrt-prod0.4

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{1 - v \cdot v}}{\color{blue}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)}}}}{t \cdot \pi}\]

  7. Applied div-inv0.4

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right) \cdot \frac{1}{1 - v \cdot v}}}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)}}}{t \cdot \pi}\]

  8. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\sqrt{2}} \cdot \frac{\frac{1}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)}}}}{t \cdot \pi}\]

  9. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\sqrt{2}}}{t} \cdot \frac{\frac{\frac{1}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(\left(-v \cdot v\right), 3, 1\right)}}}{\pi}}\]

  10. Simplified0.3

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\sqrt{2}}}{t} \cdot \color{blue}{\frac{\frac{\frac{1}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}}\]
  11. Using strategy rm

  12. Applied frac-times0.1

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\sqrt{2}} \cdot \frac{\frac{1}{\pi}}{1 - v \cdot v}}{t \cdot \sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}}\]

  13. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v \cdot v}}{\sqrt{2} \cdot \pi}}}{t \cdot \sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}\]

  14. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{2}}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot t}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))