Average Error: 0.5 → 0.3
Time: 4.8m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\sqrt{2}}}{t} \cdot \frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}\]
\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{\frac{1}{\sqrt{2}}}{t} \cdot \frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}
double f(double v, double t) {
        double r60786979 = 1.0;
        double r60786980 = 5.0;
        double r60786981 = v;
        double r60786982 = r60786981 * r60786981;
        double r60786983 = r60786980 * r60786982;
        double r60786984 = r60786979 - r60786983;
        double r60786985 = atan2(1.0, 0.0);
        double r60786986 = t;
        double r60786987 = r60786985 * r60786986;
        double r60786988 = 2.0;
        double r60786989 = 3.0;
        double r60786990 = r60786989 * r60786982;
        double r60786991 = r60786979 - r60786990;
        double r60786992 = r60786988 * r60786991;
        double r60786993 = sqrt(r60786992);
        double r60786994 = r60786987 * r60786993;
        double r60786995 = r60786979 - r60786982;
        double r60786996 = r60786994 * r60786995;
        double r60786997 = r60786984 / r60786996;
        return r60786997;
}
↓
double f(double v, double t) {
        double r60786998 = 1.0;
        double r60786999 = 2.0;
        double r60787000 = sqrt(r60786999);
        double r60787001 = r60786998 / r60787000;
        double r60787002 = t;
        double r60787003 = r60787001 / r60787002;
        double r60787004 = v;
        double r60787005 = r60787004 * r60787004;
        double r60787006 = -5.0;
        double r60787007 = fma(r60787005, r60787006, r60786998);
        double r60787008 = atan2(1.0, 0.0);
        double r60787009 = r60786998 - r60787005;
        double r60787010 = r60787008 * r60787009;
        double r60787011 = r60787007 / r60787010;
        double r60787012 = -3.0;
        double r60787013 = fma(r60787005, r60787012, r60786998);
        double r60787014 = sqrt(r60787013);
        double r60787015 = r60787011 / r60787014;
        double r60787016 = r60787003 * r60787015;
        return r60787016;
}

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. Simplified0.5

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

  4. Applied associate-/r*0.4

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

  6. Applied sqrt-prod0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Applied times-frac0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 +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)))))