Average Error: 0.4 → 0.3
Time: 22.3s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}
double f(double v, double t) {
        double r224358 = 1.0;
        double r224359 = 5.0;
        double r224360 = v;
        double r224361 = r224360 * r224360;
        double r224362 = r224359 * r224361;
        double r224363 = r224358 - r224362;
        double r224364 = atan2(1.0, 0.0);
        double r224365 = t;
        double r224366 = r224364 * r224365;
        double r224367 = 2.0;
        double r224368 = 3.0;
        double r224369 = r224368 * r224361;
        double r224370 = r224358 - r224369;
        double r224371 = r224367 * r224370;
        double r224372 = sqrt(r224371);
        double r224373 = r224366 * r224372;
        double r224374 = r224358 - r224361;
        double r224375 = r224373 * r224374;
        double r224376 = r224363 / r224375;
        return r224376;
}
↓
double f(double v, double t) {
        double r224377 = 1.0;
        double r224378 = 5.0;
        double r224379 = v;
        double r224380 = r224379 * r224379;
        double r224381 = r224378 * r224380;
        double r224382 = r224377 - r224381;
        double r224383 = t;
        double r224384 = r224382 / r224383;
        double r224385 = 1.0;
        double r224386 = atan2(1.0, 0.0);
        double r224387 = r224385 / r224386;
        double r224388 = 2.0;
        double r224389 = 3.0;
        double r224390 = r224389 * r224380;
        double r224391 = r224377 - r224390;
        double r224392 = r224388 * r224391;
        double r224393 = sqrt(r224392);
        double r224394 = r224387 / r224393;
        double r224395 = r224384 * r224394;
        double r224396 = r224377 - r224380;
        double r224397 = r224395 / r224396;
        return r224397;
}

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 associate-/r*0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied associate-/r*0.3

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

  9. Applied div-inv0.3

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

  10. Applied times-frac0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +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)))))