Average Error: 0.5 → 0.1
Time: 52.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}{t}}{1 - v \cdot v}\]
\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{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}{t}}{1 - v \cdot v}
double f(double v, double t) {
        double r6434297 = 1.0;
        double r6434298 = 5.0;
        double r6434299 = v;
        double r6434300 = r6434299 * r6434299;
        double r6434301 = r6434298 * r6434300;
        double r6434302 = r6434297 - r6434301;
        double r6434303 = atan2(1.0, 0.0);
        double r6434304 = t;
        double r6434305 = r6434303 * r6434304;
        double r6434306 = 2.0;
        double r6434307 = 3.0;
        double r6434308 = r6434307 * r6434300;
        double r6434309 = r6434297 - r6434308;
        double r6434310 = r6434306 * r6434309;
        double r6434311 = sqrt(r6434310);
        double r6434312 = r6434305 * r6434311;
        double r6434313 = r6434297 - r6434300;
        double r6434314 = r6434312 * r6434313;
        double r6434315 = r6434302 / r6434314;
        return r6434315;
}
↓
double f(double v, double t) {
        double r6434316 = -5.0;
        double r6434317 = v;
        double r6434318 = r6434317 * r6434317;
        double r6434319 = 1.0;
        double r6434320 = fma(r6434316, r6434318, r6434319);
        double r6434321 = atan2(1.0, 0.0);
        double r6434322 = r6434320 / r6434321;
        double r6434323 = -6.0;
        double r6434324 = 2.0;
        double r6434325 = fma(r6434323, r6434318, r6434324);
        double r6434326 = sqrt(r6434325);
        double r6434327 = r6434322 / r6434326;
        double r6434328 = t;
        double r6434329 = r6434327 / r6434328;
        double r6434330 = r6434319 - r6434318;
        double r6434331 = r6434329 / r6434330;
        return r6434331;
}

Error

Bits error versus v

Bits error versus t

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. Simplified0.4

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

  4. Applied div-inv0.4

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

  5. Applied times-frac0.4

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

  6. Applied associate-/l*0.4

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

  7. Simplified0.3

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

  9. Applied associate-/r*0.3

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(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)))))