Average Error: 0.4 → 0.3
Time: 32.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{1}{\pi}}{t} - \frac{\left(5 \cdot v\right) \cdot v}{t \cdot \pi}}{\left(1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\]
\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)}
↓
\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{1}{\pi}}{t} - \frac{\left(5 \cdot v\right) \cdot v}{t \cdot \pi}}{\left(1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}
double f(double v, double t) {
        double r8421547 = 1.0;
        double r8421548 = 5.0;
        double r8421549 = v;
        double r8421550 = r8421549 * r8421549;
        double r8421551 = r8421548 * r8421550;
        double r8421552 = r8421547 - r8421551;
        double r8421553 = atan2(1.0, 0.0);
        double r8421554 = t;
        double r8421555 = r8421553 * r8421554;
        double r8421556 = 2.0;
        double r8421557 = 3.0;
        double r8421558 = r8421557 * r8421550;
        double r8421559 = r8421547 - r8421558;
        double r8421560 = r8421556 * r8421559;
        double r8421561 = sqrt(r8421560);
        double r8421562 = r8421555 * r8421561;
        double r8421563 = r8421547 - r8421550;
        double r8421564 = r8421562 * r8421563;
        double r8421565 = r8421552 / r8421564;
        return r8421565;
}
↓
double f(double v, double t) {
        double r8421566 = 1.0;
        double r8421567 = r8421566 * r8421566;
        double r8421568 = v;
        double r8421569 = r8421568 * r8421568;
        double r8421570 = r8421569 * r8421566;
        double r8421571 = r8421569 * r8421569;
        double r8421572 = r8421570 + r8421571;
        double r8421573 = r8421567 + r8421572;
        double r8421574 = atan2(1.0, 0.0);
        double r8421575 = r8421566 / r8421574;
        double r8421576 = t;
        double r8421577 = r8421575 / r8421576;
        double r8421578 = 5.0;
        double r8421579 = r8421578 * r8421568;
        double r8421580 = r8421579 * r8421568;
        double r8421581 = r8421576 * r8421574;
        double r8421582 = r8421580 / r8421581;
        double r8421583 = r8421577 - r8421582;
        double r8421584 = r8421566 * r8421567;
        double r8421585 = r8421569 * r8421571;
        double r8421586 = r8421584 - r8421585;
        double r8421587 = 2.0;
        double r8421588 = 3.0;
        double r8421589 = r8421588 * r8421569;
        double r8421590 = r8421566 - r8421589;
        double r8421591 = r8421587 * r8421590;
        double r8421592 = sqrt(r8421591);
        double r8421593 = r8421586 * r8421592;
        double r8421594 = r8421583 / r8421593;
        double r8421595 = r8421573 * r8421594;
        return r8421595;
}

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 flip3--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied associate-/r/0.4

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

  6. Simplified0.4

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

  7. Taylor expanded around 0 0.4

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))