Average Error: 0.4 → 0.6
Time: 24.7s
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 r185312 = 1.0;
        double r185313 = 5.0;
        double r185314 = v;
        double r185315 = r185314 * r185314;
        double r185316 = r185313 * r185315;
        double r185317 = r185312 - r185316;
        double r185318 = atan2(1.0, 0.0);
        double r185319 = t;
        double r185320 = r185318 * r185319;
        double r185321 = 2.0;
        double r185322 = 3.0;
        double r185323 = r185322 * r185315;
        double r185324 = r185312 - r185323;
        double r185325 = r185321 * r185324;
        double r185326 = sqrt(r185325);
        double r185327 = r185320 * r185326;
        double r185328 = r185312 - r185315;
        double r185329 = r185327 * r185328;
        double r185330 = r185317 / r185329;
        return r185330;
}
↓
double f(double v, double t) {
        double r185331 = 1.0;
        double r185332 = sqrt(r185331);
        double r185333 = t;
        double r185334 = 2.0;
        double r185335 = sqrt(r185334);
        double r185336 = atan2(1.0, 0.0);
        double r185337 = r185335 * r185336;
        double r185338 = r185333 * r185337;
        double r185339 = r185332 / r185338;
        double r185340 = r185331 * r185339;
        double r185341 = r185332 / r185337;
        double r185342 = 4.0;
        double r185343 = r185341 * r185342;
        double r185344 = v;
        double r185345 = 2.0;
        double r185346 = pow(r185344, r185345);
        double r185347 = r185346 / r185333;
        double r185348 = 4.0;
        double r185349 = pow(r185344, r185348);
        double r185350 = r185349 / r185333;
        double r185351 = r185347 + r185350;
        double r185352 = r185343 * r185351;
        double r185353 = 1.125;
        double r185354 = 3.0;
        double r185355 = pow(r185332, r185354);
        double r185356 = r185355 * r185336;
        double r185357 = r185335 * r185356;
        double r185358 = r185353 / r185357;
        double r185359 = 1.5;
        double r185360 = r185332 * r185336;
        double r185361 = r185335 * r185360;
        double r185362 = r185359 / r185361;
        double r185363 = r185358 + r185362;
        double r185364 = r185350 * r185363;
        double r185365 = r185333 * r185361;
        double r185366 = r185346 / r185365;
        double r185367 = r185359 * r185366;
        double r185368 = r185364 - r185367;
        double r185369 = r185352 + r185368;
        double r185370 = -r185369;
        double r185371 = r185340 + r185370;
        return r185371;
}

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. 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 2019297 
(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)))))