Average Error: 0.4 → 0.1
Time: 8.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(5 \cdot v, v, 1\right)}{\mathsf{fma}\left(5 \cdot v, v, 1\right) \cdot \pi} \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}}{t}\]
\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(5 \cdot v, v, 1\right)}{\mathsf{fma}\left(5 \cdot v, v, 1\right) \cdot \pi} \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}}{t}
double f(double v, double t) {
        double r338795 = 1.0;
        double r338796 = 5.0;
        double r338797 = v;
        double r338798 = r338797 * r338797;
        double r338799 = r338796 * r338798;
        double r338800 = r338795 - r338799;
        double r338801 = atan2(1.0, 0.0);
        double r338802 = t;
        double r338803 = r338801 * r338802;
        double r338804 = 2.0;
        double r338805 = 3.0;
        double r338806 = r338805 * r338798;
        double r338807 = r338795 - r338806;
        double r338808 = r338804 * r338807;
        double r338809 = sqrt(r338808);
        double r338810 = r338803 * r338809;
        double r338811 = r338795 - r338798;
        double r338812 = r338810 * r338811;
        double r338813 = r338800 / r338812;
        return r338813;
}
↓
double f(double v, double t) {
        double r338814 = 5.0;
        double r338815 = v;
        double r338816 = r338814 * r338815;
        double r338817 = 1.0;
        double r338818 = fma(r338816, r338815, r338817);
        double r338819 = atan2(1.0, 0.0);
        double r338820 = r338818 * r338819;
        double r338821 = r338818 / r338820;
        double r338822 = r338815 * r338815;
        double r338823 = r338814 * r338822;
        double r338824 = r338817 - r338823;
        double r338825 = r338821 * r338824;
        double r338826 = 2.0;
        double r338827 = 3.0;
        double r338828 = r338827 * r338822;
        double r338829 = r338817 - r338828;
        double r338830 = r338826 * r338829;
        double r338831 = sqrt(r338830);
        double r338832 = r338817 - r338822;
        double r338833 = r338831 * r338832;
        double r338834 = r338825 / r338833;
        double r338835 = t;
        double r338836 = r338834 / r338835;
        return r338836;
}

Error

Bits error versus v

Bits error versus t

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 flip--0.4

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

  4. Applied associate-/l/0.4

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

  5. Simplified0.4

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

  7. Applied associate-/r*0.4

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

  8. Simplified0.4

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

  10. Applied associate-*r/0.3

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

  11. Applied associate-/l/0.3

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

  13. Applied associate-/r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))))