Average Error: 0.6 → 0.6
Time: 11.5s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\ {\left({e}^{t_0}\right)}^{t_0} \end{array} \]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\begin{array}{l}
t_0 := \sqrt{\log \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\\
{\left({e}^{t_0}\right)}^{t_0}
\end{array}

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (log (acos (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))))
   (pow (pow E t_0) t_0)))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	double t_0 = sqrt(log(acos(fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0))));
	return pow(pow(((double) M_E), t_0), t_0);
}

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

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

  3. Applied add-exp-log_binary640.6

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

  4. Applied add-sqr-sqrt_binary640.6

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

  5. Applied exp-prod_binary640.6

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

  6. Applied pow1_binary640.6

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

  7. Applied log-pow_binary640.6

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

  8. Applied sqrt-prod_binary640.6

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

  9. Applied exp-prod_binary640.6

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2022082 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))