Average Error: 0.0 → 0.0
Time: 6.6s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{{\left(1 - x\right)}^{3}}{{\left(x + 1\right)}^{3}}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{{\left(1 - x\right)}^{3}}{{\left(x + 1\right)}^{3}}}}\right)
double f(double x) {
        double r11548 = 2.0;
        double r11549 = 1.0;
        double r11550 = x;
        double r11551 = r11549 - r11550;
        double r11552 = r11549 + r11550;
        double r11553 = r11551 / r11552;
        double r11554 = sqrt(r11553);
        double r11555 = atan(r11554);
        double r11556 = r11548 * r11555;
        return r11556;
}
↓
double f(double x) {
        double r11557 = 2.0;
        double r11558 = 1.0;
        double r11559 = x;
        double r11560 = r11558 - r11559;
        double r11561 = 3.0;
        double r11562 = pow(r11560, r11561);
        double r11563 = r11559 + r11558;
        double r11564 = pow(r11563, r11561);
        double r11565 = r11562 / r11564;
        double r11566 = cbrt(r11565);
        double r11567 = sqrt(r11566);
        double r11568 = atan(r11567);
        double r11569 = r11557 * r11568;
        return r11569;
}

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. Using strategy rm

  3. Applied add-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied cbrt-undiv0.0

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

  6. Simplified0.0

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

  8. Applied cube-div0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))