Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{x + 1}\right)}^{3}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{x + 1}\right)}^{3}}}\right) \cdot 2
double f(double x) {
        double r18648 = 2.0;
        double r18649 = 1.0;
        double r18650 = x;
        double r18651 = r18649 - r18650;
        double r18652 = r18649 + r18650;
        double r18653 = r18651 / r18652;
        double r18654 = sqrt(r18653);
        double r18655 = atan(r18654);
        double r18656 = r18648 * r18655;
        return r18656;
}
↓
double f(double x) {
        double r18657 = 1.0;
        double r18658 = x;
        double r18659 = r18657 - r18658;
        double r18660 = 1.0;
        double r18661 = r18658 + r18657;
        double r18662 = r18660 / r18661;
        double r18663 = r18659 * r18662;
        double r18664 = 3.0;
        double r18665 = pow(r18663, r18664);
        double r18666 = cbrt(r18665);
        double r18667 = sqrt(r18666);
        double r18668 = atan(r18667);
        double r18669 = 2.0;
        double r18670 = r18668 * r18669;
        return r18670;
}

Error

Bits error versus x

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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. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

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

  9. Applied div-inv0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))