Average Error: 0.4 → 0.3
Time: 7.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}}{\frac{1}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}\]
\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{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}}{\frac{1}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}
double f(double v, double t) {
        double r344609 = 1.0;
        double r344610 = 5.0;
        double r344611 = v;
        double r344612 = r344611 * r344611;
        double r344613 = r344610 * r344612;
        double r344614 = r344609 - r344613;
        double r344615 = atan2(1.0, 0.0);
        double r344616 = t;
        double r344617 = r344615 * r344616;
        double r344618 = 2.0;
        double r344619 = 3.0;
        double r344620 = r344619 * r344612;
        double r344621 = r344609 - r344620;
        double r344622 = r344618 * r344621;
        double r344623 = sqrt(r344622);
        double r344624 = r344617 * r344623;
        double r344625 = r344609 - r344612;
        double r344626 = r344624 * r344625;
        double r344627 = r344614 / r344626;
        return r344627;
}
↓
double f(double v, double t) {
        double r344628 = 1.0;
        double r344629 = 5.0;
        double r344630 = v;
        double r344631 = r344630 * r344630;
        double r344632 = r344629 * r344631;
        double r344633 = r344628 - r344632;
        double r344634 = sqrt(r344633);
        double r344635 = atan2(1.0, 0.0);
        double r344636 = r344634 / r344635;
        double r344637 = t;
        double r344638 = r344636 / r344637;
        double r344639 = 2.0;
        double r344640 = 3.0;
        double r344641 = r344640 * r344631;
        double r344642 = r344628 - r344641;
        double r344643 = r344639 * r344642;
        double r344644 = sqrt(r344643);
        double r344645 = r344628 - r344631;
        double r344646 = r344644 * r344645;
        double r344647 = r344638 / r344646;
        double r344648 = 1.0;
        double r344649 = r344648 / r344634;
        double r344650 = r344647 / r344649;
        return r344650;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied associate-/l*0.4

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

  6. Applied add-log-exp0.4

    \[\leadsto \frac{\sqrt{1 - \color{blue}{\log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}}}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}\]
  7. Using strategy rm

  8. Applied div-inv0.4

    \[\leadsto \frac{\sqrt{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}}{\color{blue}{\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \frac{1}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}}\]

  9. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}}{\frac{1}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}}\]

  10. Simplified0.4

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

  12. Applied associate-/r*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020059 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))