Average Error: 0.0 → 0.0
Time: 12.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1}{\left(1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)} - \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)}{\left(\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)} + \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \frac{\frac{x}{1 + x}}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1}{\left(1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)} - \frac{x}{1 + x} \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)}{\left(\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)} + \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \frac{\frac{x}{1 + x}}{1 + x}}}\right)
double f(double x) {
        double r315997 = 2.0;
        double r315998 = 1.0;
        double r315999 = x;
        double r316000 = r315998 - r315999;
        double r316001 = r315998 + r315999;
        double r316002 = r316000 / r316001;
        double r316003 = sqrt(r316002);
        double r316004 = atan(r316003);
        double r316005 = r315997 * r316004;
        return r316005;
}
↓
double f(double x) {
        double r316006 = 2.0;
        double r316007 = 1.0;
        double r316008 = x;
        double r316009 = r316007 + r316008;
        double r316010 = r316009 * r316009;
        double r316011 = r316009 * r316010;
        double r316012 = r316007 / r316011;
        double r316013 = r316008 / r316009;
        double r316014 = r316013 * r316013;
        double r316015 = r316013 * r316014;
        double r316016 = r316012 - r316015;
        double r316017 = r316007 / r316010;
        double r316018 = r316017 + r316014;
        double r316019 = r316013 / r316009;
        double r316020 = r316018 + r316019;
        double r316021 = r316016 / r316020;
        double r316022 = sqrt(r316021);
        double r316023 = atan(r316022);
        double r316024 = r316006 * r316023;
        return r316024;
}

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 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 flip3--0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))