Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{x + 1}\right)}^{3}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{x + 1}\right)}^{3}}}\right) \cdot 2
double f(double x) {
        double r13127 = 2.0;
        double r13128 = 1.0;
        double r13129 = x;
        double r13130 = r13128 - r13129;
        double r13131 = r13128 + r13129;
        double r13132 = r13130 / r13131;
        double r13133 = sqrt(r13132);
        double r13134 = atan(r13133);
        double r13135 = r13127 * r13134;
        return r13135;
}
↓
double f(double x) {
        double r13136 = 1.0;
        double r13137 = x;
        double r13138 = r13136 - r13137;
        double r13139 = r13137 + r13136;
        double r13140 = r13138 / r13139;
        double r13141 = 3.0;
        double r13142 = pow(r13140, r13141);
        double r13143 = cbrt(r13142);
        double r13144 = sqrt(r13143);
        double r13145 = atan(r13144);
        double r13146 = 2.0;
        double r13147 = r13145 * r13146;
        return r13147;
}

Error

Bits error versus x

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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. 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

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))