Average Error: 0.0 → 0.0
Time: 1.7m
Precision: 64
Internal Precision: 384
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{\frac{1 - x}{\sqrt{x + 1}} \cdot \left(\frac{1 - x}{x + 1} \cdot \left(1 - x\right)\right)}}{\sqrt{1 + x}}}\right)\]

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-sqr-sqrt0.0

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

  4. Applied associate-/r*0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Applied cbrt-undiv0.0

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

  9. Applied simplify0.0

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))