Average Error: 0.5 → 0.3
Time: 49.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}, v \cdot \left(-v\right), \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right)}\]
\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{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}, v \cdot \left(-v\right), \sqrt{\mathsf{fma}\left(6, v \cdot \left(-v\right), 2\right)}\right)}
double f(double v, double t) {
        double r11061986 = 1.0;
        double r11061987 = 5.0;
        double r11061988 = v;
        double r11061989 = r11061988 * r11061988;
        double r11061990 = r11061987 * r11061989;
        double r11061991 = r11061986 - r11061990;
        double r11061992 = atan2(1.0, 0.0);
        double r11061993 = t;
        double r11061994 = r11061992 * r11061993;
        double r11061995 = 2.0;
        double r11061996 = 3.0;
        double r11061997 = r11061996 * r11061989;
        double r11061998 = r11061986 - r11061997;
        double r11061999 = r11061995 * r11061998;
        double r11062000 = sqrt(r11061999);
        double r11062001 = r11061994 * r11062000;
        double r11062002 = r11061986 - r11061989;
        double r11062003 = r11062001 * r11062002;
        double r11062004 = r11061991 / r11062003;
        return r11062004;
}
↓
double f(double v, double t) {
        double r11062005 = v;
        double r11062006 = r11062005 * r11062005;
        double r11062007 = -5.0;
        double r11062008 = 1.0;
        double r11062009 = fma(r11062006, r11062007, r11062008);
        double r11062010 = atan2(1.0, 0.0);
        double r11062011 = r11062009 / r11062010;
        double r11062012 = t;
        double r11062013 = r11062011 / r11062012;
        double r11062014 = 6.0;
        double r11062015 = -r11062005;
        double r11062016 = r11062005 * r11062015;
        double r11062017 = 2.0;
        double r11062018 = fma(r11062014, r11062016, r11062017);
        double r11062019 = sqrt(r11062018);
        double r11062020 = fma(r11062019, r11062016, r11062019);
        double r11062021 = r11062013 / r11062020;
        return r11062021;
}

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. Simplified0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 +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)))))