Average Error: 0.4 → 0.3
Time: 21.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}
double f(double v, double t) {
        double r278350 = 1.0;
        double r278351 = 5.0;
        double r278352 = v;
        double r278353 = r278352 * r278352;
        double r278354 = r278351 * r278353;
        double r278355 = r278350 - r278354;
        double r278356 = atan2(1.0, 0.0);
        double r278357 = t;
        double r278358 = r278356 * r278357;
        double r278359 = 2.0;
        double r278360 = 3.0;
        double r278361 = r278360 * r278353;
        double r278362 = r278350 - r278361;
        double r278363 = r278359 * r278362;
        double r278364 = sqrt(r278363);
        double r278365 = r278358 * r278364;
        double r278366 = r278350 - r278353;
        double r278367 = r278365 * r278366;
        double r278368 = r278355 / r278367;
        return r278368;
}
↓
double f(double v, double t) {
        double r278369 = 1.0;
        double r278370 = 5.0;
        double r278371 = v;
        double r278372 = r278371 * r278371;
        double r278373 = r278370 * r278372;
        double r278374 = r278369 - r278373;
        double r278375 = atan2(1.0, 0.0);
        double r278376 = r278374 / r278375;
        double r278377 = t;
        double r278378 = 2.0;
        double r278379 = 3.0;
        double r278380 = r278379 * r278372;
        double r278381 = r278369 - r278380;
        double r278382 = r278378 * r278381;
        double r278383 = sqrt(r278382);
        double r278384 = r278377 * r278383;
        double r278385 = r278376 / r278384;
        double r278386 = r278369 - r278372;
        double r278387 = r278385 / r278386;
        return r278387;
}

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 associate-/r*0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied associate-/r*0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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)))))