Average Error: 0.0 → 0.0
Time: 53.8s
Precision: 64
Internal Precision: 128
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{\sqrt{\left(1 + x\right) \cdot \left(1 + x\right)}}\right)\]

Error

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Results

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Derivation

  1. Initial program 0.0

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

  3. Applied div-sub0.0

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

  5. Applied frac-sub0.0

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

  6. Applied sqrt-div0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Runtime

Time bar (total: 53.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.00.00.00.0100%
herbie shell --seed 2018354 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))