Average Error: 0.0 → 0.0
Time: 10.2s
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 r18531 = 2.0;
        double r18532 = 1.0;
        double r18533 = x;
        double r18534 = r18532 - r18533;
        double r18535 = r18532 + r18533;
        double r18536 = r18534 / r18535;
        double r18537 = sqrt(r18536);
        double r18538 = atan(r18537);
        double r18539 = r18531 * r18538;
        return r18539;
}
↓
double f(double x) {
        double r18540 = 1.0;
        double r18541 = x;
        double r18542 = r18540 - r18541;
        double r18543 = 1.0;
        double r18544 = r18541 + r18540;
        double r18545 = r18543 / r18544;
        double r18546 = r18542 * r18545;
        double r18547 = 3.0;
        double r18548 = pow(r18546, r18547);
        double r18549 = cbrt(r18548);
        double r18550 = sqrt(r18549);
        double r18551 = atan(r18550);
        double r18552 = 2.0;
        double r18553 = r18551 * r18552;
        return r18553;
}

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. 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. Using strategy rm

  9. Applied div-inv0.0

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

  10. 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 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))