Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right)}\right)
double f(double x) {
        double r15038 = 2.0;
        double r15039 = 1.0;
        double r15040 = x;
        double r15041 = r15039 - r15040;
        double r15042 = r15039 + r15040;
        double r15043 = r15041 / r15042;
        double r15044 = sqrt(r15043);
        double r15045 = atan(r15044);
        double r15046 = r15038 * r15045;
        return r15046;
}
↓
double f(double x) {
        double r15047 = 2.0;
        double r15048 = 1.0;
        double r15049 = x;
        double r15050 = r15048 - r15049;
        double r15051 = r15048 * r15048;
        double r15052 = r15049 * r15049;
        double r15053 = r15051 - r15052;
        double r15054 = r15050 / r15053;
        double r15055 = r15054 * r15050;
        double r15056 = sqrt(r15055);
        double r15057 = atan(r15056);
        double r15058 = r15047 * r15057;
        return r15058;
}

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 flip-+0.0

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

  4. Applied associate-/r/0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))