Average Error: 0.6 → 0.6
Time: 29.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left({e}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left({e}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}
double f(double v) {
        double r7404987 = 1.0;
        double r7404988 = 5.0;
        double r7404989 = v;
        double r7404990 = r7404989 * r7404989;
        double r7404991 = r7404988 * r7404990;
        double r7404992 = r7404987 - r7404991;
        double r7404993 = r7404990 - r7404987;
        double r7404994 = r7404992 / r7404993;
        double r7404995 = acos(r7404994);
        return r7404995;
}
↓
double f(double v) {
        double r7404996 = exp(1.0);
        double r7404997 = v;
        double r7404998 = -5.0;
        double r7404999 = r7404997 * r7404998;
        double r7405000 = 1.0;
        double r7405001 = fma(r7404999, r7404997, r7405000);
        double r7405002 = -1.0;
        double r7405003 = fma(r7404997, r7404997, r7405002);
        double r7405004 = r7405001 / r7405003;
        double r7405005 = acos(r7405004);
        double r7405006 = log(r7405005);
        double r7405007 = sqrt(r7405006);
        double r7405008 = pow(r7404996, r7405007);
        double r7405009 = pow(r7405008, r7405007);
        return r7405009;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

  2. Simplified0.6

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

  4. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.6

    \[\leadsto e^{\color{blue}{\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.6

    \[\leadsto e^{\color{blue}{\log \left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\]

  9. Applied exp-to-pow0.6

    \[\leadsto \color{blue}{{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.6

    \[\leadsto {\left(e^{\sqrt{\color{blue}{1 \cdot \log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]

  12. Applied sqrt-prod0.6

    \[\leadsto {\left(e^{\color{blue}{\sqrt{1} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]

  13. Applied exp-prod0.6

    \[\leadsto {\color{blue}{\left({\left(e^{\sqrt{1}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]

  14. Simplified0.6

    \[\leadsto {\left({\color{blue}{e}}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]

  15. Final simplification0.6

    \[\leadsto {\left({e}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))