Average Error: 0.5 → 0.1
Time: 1.5m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1}{\pi} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{t}}{1 - v \cdot v}\]
\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{\frac{1}{\pi} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{t}}{1 - v \cdot v}
double f(double v, double t) {
        double r36039277 = 1.0;
        double r36039278 = 5.0;
        double r36039279 = v;
        double r36039280 = r36039279 * r36039279;
        double r36039281 = r36039278 * r36039280;
        double r36039282 = r36039277 - r36039281;
        double r36039283 = atan2(1.0, 0.0);
        double r36039284 = t;
        double r36039285 = r36039283 * r36039284;
        double r36039286 = 2.0;
        double r36039287 = 3.0;
        double r36039288 = r36039287 * r36039280;
        double r36039289 = r36039277 - r36039288;
        double r36039290 = r36039286 * r36039289;
        double r36039291 = sqrt(r36039290);
        double r36039292 = r36039285 * r36039291;
        double r36039293 = r36039277 - r36039280;
        double r36039294 = r36039292 * r36039293;
        double r36039295 = r36039282 / r36039294;
        return r36039295;
}
↓
double f(double v, double t) {
        double r36039296 = 1.0;
        double r36039297 = atan2(1.0, 0.0);
        double r36039298 = r36039296 / r36039297;
        double r36039299 = v;
        double r36039300 = r36039299 * r36039299;
        double r36039301 = 5.0;
        double r36039302 = r36039300 * r36039301;
        double r36039303 = r36039296 - r36039302;
        double r36039304 = 2.0;
        double r36039305 = -6.0;
        double r36039306 = r36039300 * r36039305;
        double r36039307 = r36039304 + r36039306;
        double r36039308 = sqrt(r36039307);
        double r36039309 = r36039303 / r36039308;
        double r36039310 = r36039298 * r36039309;
        double r36039311 = t;
        double r36039312 = r36039310 / r36039311;
        double r36039313 = r36039296 - r36039300;
        double r36039314 = r36039312 / r36039313;
        return r36039314;
}

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 associate-/r*0.5

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

  5. Applied *-un-lft-identity0.5

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

  9. Applied associate-/r*0.3

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

  11. Applied associate-*l/0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019120 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))