Average Error: 0.4 → 0.5
Time: 23.0s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)\right)\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right)\right)\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r250428 = 1.0;
        double r250429 = 5.0;
        double r250430 = v;
        double r250431 = r250430 * r250430;
        double r250432 = r250429 * r250431;
        double r250433 = r250428 - r250432;
        double r250434 = atan2(1.0, 0.0);
        double r250435 = t;
        double r250436 = r250434 * r250435;
        double r250437 = 2.0;
        double r250438 = 3.0;
        double r250439 = r250438 * r250431;
        double r250440 = r250428 - r250439;
        double r250441 = r250437 * r250440;
        double r250442 = sqrt(r250441);
        double r250443 = r250436 * r250442;
        double r250444 = r250428 - r250431;
        double r250445 = r250443 * r250444;
        double r250446 = r250433 / r250445;
        return r250446;
}
↓
double f(double v, double t) {
        double r250447 = 1.0;
        double r250448 = 5.0;
        double r250449 = v;
        double r250450 = r250449 * r250449;
        double r250451 = r250448 * r250450;
        double r250452 = r250447 - r250451;
        double r250453 = atan2(1.0, 0.0);
        double r250454 = t;
        double r250455 = 2.0;
        double r250456 = 3.0;
        double r250457 = r250456 * r250450;
        double r250458 = r250447 - r250457;
        double r250459 = r250455 * r250458;
        double r250460 = sqrt(r250459);
        double r250461 = cbrt(r250460);
        double r250462 = r250461 * r250461;
        double r250463 = r250454 * r250462;
        double r250464 = r250453 * r250463;
        double r250465 = r250464 * r250461;
        double r250466 = r250447 - r250450;
        double r250467 = r250465 * r250466;
        double r250468 = r250452 / r250467;
        return r250468;
}

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-cube-cbrt0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied associate-*l*0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019347 +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)))))