Average Error: 0.5 → 0.6
Time: 25.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \left(-\left(\left(\frac{\sqrt{1}}{\sqrt{2} \cdot \pi} \cdot 4\right) \cdot \left(\frac{{v}^{2}}{t} + \frac{{v}^{4}}{t}\right) + \left(\frac{{v}^{4}}{t} \cdot \left(\frac{1.125}{\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)} + \frac{1.5}{\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)}\right) - 1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}\right)\right)\right)\]
\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)}
↓
1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \left(-\left(\left(\frac{\sqrt{1}}{\sqrt{2} \cdot \pi} \cdot 4\right) \cdot \left(\frac{{v}^{2}}{t} + \frac{{v}^{4}}{t}\right) + \left(\frac{{v}^{4}}{t} \cdot \left(\frac{1.125}{\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)} + \frac{1.5}{\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)}\right) - 1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}\right)\right)\right)
double f(double v, double t) {
        double r194330 = 1.0;
        double r194331 = 5.0;
        double r194332 = v;
        double r194333 = r194332 * r194332;
        double r194334 = r194331 * r194333;
        double r194335 = r194330 - r194334;
        double r194336 = atan2(1.0, 0.0);
        double r194337 = t;
        double r194338 = r194336 * r194337;
        double r194339 = 2.0;
        double r194340 = 3.0;
        double r194341 = r194340 * r194333;
        double r194342 = r194330 - r194341;
        double r194343 = r194339 * r194342;
        double r194344 = sqrt(r194343);
        double r194345 = r194338 * r194344;
        double r194346 = r194330 - r194333;
        double r194347 = r194345 * r194346;
        double r194348 = r194335 / r194347;
        return r194348;
}
↓
double f(double v, double t) {
        double r194349 = 1.0;
        double r194350 = sqrt(r194349);
        double r194351 = t;
        double r194352 = 2.0;
        double r194353 = sqrt(r194352);
        double r194354 = atan2(1.0, 0.0);
        double r194355 = r194353 * r194354;
        double r194356 = r194351 * r194355;
        double r194357 = r194350 / r194356;
        double r194358 = r194349 * r194357;
        double r194359 = r194350 / r194355;
        double r194360 = 4.0;
        double r194361 = r194359 * r194360;
        double r194362 = v;
        double r194363 = 2.0;
        double r194364 = pow(r194362, r194363);
        double r194365 = r194364 / r194351;
        double r194366 = 4.0;
        double r194367 = pow(r194362, r194366);
        double r194368 = r194367 / r194351;
        double r194369 = r194365 + r194368;
        double r194370 = r194361 * r194369;
        double r194371 = 1.125;
        double r194372 = 3.0;
        double r194373 = pow(r194350, r194372);
        double r194374 = r194373 * r194354;
        double r194375 = r194353 * r194374;
        double r194376 = r194371 / r194375;
        double r194377 = 1.5;
        double r194378 = r194350 * r194354;
        double r194379 = r194353 * r194378;
        double r194380 = r194377 / r194379;
        double r194381 = r194376 + r194380;
        double r194382 = r194368 * r194381;
        double r194383 = r194351 * r194379;
        double r194384 = r194364 / r194383;
        double r194385 = r194377 * r194384;
        double r194386 = r194382 - r194385;
        double r194387 = r194370 + r194386;
        double r194388 = -r194387;
        double r194389 = r194358 + r194388;
        return r194389;
}

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. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)} + \left(4 \cdot \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}\right) + 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \left(-\left(\left(\frac{\sqrt{1}}{\sqrt{2} \cdot \pi} \cdot 4\right) \cdot \left(\frac{{v}^{2}}{t} + \frac{{v}^{4}}{t}\right) + \left(\frac{{v}^{4}}{t} \cdot \left(\frac{1.125}{\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)} + \frac{1.5}{\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)}\right) - 1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019298 
(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)))))