Average Error: 0.0 → 0.0
Time: 1.5m
Precision: 64
Internal Precision: 320
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{\log \left(1 - x\right) - \log_* (1 + x)}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied add-exp-log0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{e^{\log \left(1 + x\right)}}}}\right)\]

  4. Applied add-exp-log0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{e^{\log \left(1 - x\right)}}}{e^{\log \left(1 + x\right)}}}\right)\]

  5. Applied div-exp0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{e^{\log \left(1 - x\right) - \log \left(1 + x\right)}}}\right)\]

  6. Applied simplify0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{e^{\color{blue}{\log \left(1 - x\right) - \log_* (1 + x)}}}\right)\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))