Average Error: 0.0 → 0.0
Time: 48.4s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{(e^{\log_* (1 + \sqrt[3]{\frac{1 - x}{x + 1}} \cdot \sqrt[3]{\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}})} - 1)^*}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Initial simplification0.0

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

  4. Applied expm1-log1p-u0.0

    \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}}\right) \cdot 2\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.0

    \[\leadsto \tan^{-1} \left(\sqrt{(e^{\log_* (1 + \color{blue}{\sqrt[3]{\left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right) \cdot \frac{1 - x}{1 + x}}})} - 1)^*}\right) \cdot 2\]
  7. Using strategy rm

  8. Applied cbrt-prod0.0

    \[\leadsto \tan^{-1} \left(\sqrt{(e^{\log_* (1 + \color{blue}{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}})} - 1)^*}\right) \cdot 2\]

  9. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{(e^{\log_* (1 + \sqrt[3]{\frac{1 - x}{x + 1}} \cdot \sqrt[3]{\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}})} - 1)^*}\right)\]

Runtime

Time bar (total: 48.4s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))