Average Error: 0.4 → 0.4
Time: 8.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(1 + v \cdot v\right)\]
\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 t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(1 + v \cdot v\right)
double f(double v, double t) {
        double r196374 = 1.0;
        double r196375 = 5.0;
        double r196376 = v;
        double r196377 = r196376 * r196376;
        double r196378 = r196375 * r196377;
        double r196379 = r196374 - r196378;
        double r196380 = atan2(1.0, 0.0);
        double r196381 = t;
        double r196382 = r196380 * r196381;
        double r196383 = 2.0;
        double r196384 = 3.0;
        double r196385 = r196384 * r196377;
        double r196386 = r196374 - r196385;
        double r196387 = r196383 * r196386;
        double r196388 = sqrt(r196387);
        double r196389 = r196382 * r196388;
        double r196390 = r196374 - r196377;
        double r196391 = r196389 * r196390;
        double r196392 = r196379 / r196391;
        return r196392;
}
↓
double f(double v, double t) {
        double r196393 = 1.0;
        double r196394 = 5.0;
        double r196395 = v;
        double r196396 = r196395 * r196395;
        double r196397 = r196394 * r196396;
        double r196398 = r196393 - r196397;
        double r196399 = atan2(1.0, 0.0);
        double r196400 = t;
        double r196401 = r196399 * r196400;
        double r196402 = 2.0;
        double r196403 = 3.0;
        double r196404 = r196403 * r196396;
        double r196405 = r196393 - r196404;
        double r196406 = r196402 * r196405;
        double r196407 = sqrt(r196406);
        double r196408 = r196401 * r196407;
        double r196409 = r196393 * r196393;
        double r196410 = r196396 * r196396;
        double r196411 = r196409 - r196410;
        double r196412 = r196408 * r196411;
        double r196413 = r196398 / r196412;
        double r196414 = r196393 + r196396;
        double r196415 = r196413 * r196414;
        return r196415;
}

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 flip--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied associate-/r/0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 
(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)))))