Average Error: 0.6 → 0.6
Time: 2.3m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;v \cdot v \le 6.4761694066342 \cdot 10^{-34}:\\ \;\;\;\;\cos^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + 1}}{v - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\\ \end{array}\]

Error

Bits error versus v

Derivation

  1. Split input into 2 regimes

  2. if (* v v) < 6.4761694066342e-34

    1. Initial program 0

      \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
    2. Using strategy rm

    3. Applied *-commutative0

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

    4. Applied difference-of-sqr-10

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

    5. Applied associate-/r*0

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

    6. Taylor expanded around inf 0

      \[\leadsto \cos^{-1} \left(\frac{\frac{\color{blue}{1 - 5 \cdot {v}^{2}}}{v + 1}}{v - 1}\right)\]

    7. Simplified0

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

    if 6.4761694066342e-34 < (* v v)

    1. Initial program 9.8

      \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;v \cdot v \le 6.4761694066342 \cdot 10^{-34}:\\ \;\;\;\;\cos^{-1} \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v + 1}}{v - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))