Average Error: 0.5 → 0.3
Time: 13.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{2} \cdot t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]
\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}}{\sqrt{2} \cdot t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\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)
double f(double v, double t) {
        double r183326 = 1.0;
        double r183327 = 5.0;
        double r183328 = v;
        double r183329 = r183328 * r183328;
        double r183330 = r183327 * r183329;
        double r183331 = r183326 - r183330;
        double r183332 = atan2(1.0, 0.0);
        double r183333 = t;
        double r183334 = r183332 * r183333;
        double r183335 = 2.0;
        double r183336 = 3.0;
        double r183337 = r183336 * r183329;
        double r183338 = r183326 - r183337;
        double r183339 = r183335 * r183338;
        double r183340 = sqrt(r183339);
        double r183341 = r183334 * r183340;
        double r183342 = r183326 - r183329;
        double r183343 = r183341 * r183342;
        double r183344 = r183331 / r183343;
        return r183344;
}
↓
double f(double v, double t) {
        double r183345 = 1.0;
        double r183346 = 5.0;
        double r183347 = v;
        double r183348 = r183347 * r183347;
        double r183349 = r183346 * r183348;
        double r183350 = r183345 - r183349;
        double r183351 = atan2(1.0, 0.0);
        double r183352 = r183350 / r183351;
        double r183353 = 2.0;
        double r183354 = sqrt(r183353);
        double r183355 = t;
        double r183356 = r183354 * r183355;
        double r183357 = r183352 / r183356;
        double r183358 = 3.0;
        double r183359 = r183358 * r183348;
        double r183360 = r183345 - r183359;
        double r183361 = sqrt(r183360);
        double r183362 = 3.0;
        double r183363 = pow(r183345, r183362);
        double r183364 = 6.0;
        double r183365 = pow(r183347, r183364);
        double r183366 = r183363 - r183365;
        double r183367 = r183361 * r183366;
        double r183368 = r183357 / r183367;
        double r183369 = r183345 * r183345;
        double r183370 = r183348 * r183348;
        double r183371 = r183345 * r183348;
        double r183372 = r183370 + r183371;
        double r183373 = r183369 + r183372;
        double r183374 = r183368 * r183373;
        return r183374;
}

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 0.5

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

  3. Applied sqrt-prod0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied associate-*r*0.5

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

  9. Applied flip3--0.5

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

  10. Applied associate-*r/0.5

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

  11. Applied associate-/r/0.5

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

  12. Simplified0.4

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

  13. Taylor expanded around 0 0.4

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

  14. Simplified0.3

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

  15. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{2} \cdot t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} - {v}^{6}\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

Reproduce

herbie shell --seed 2020046 
(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)))))