Average Error: 0.0 → 0.0
Time: 6.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\left|\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right|\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\left|\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right|\right)
double f(double x) {
        double r9986 = 2.0;
        double r9987 = 1.0;
        double r9988 = x;
        double r9989 = r9987 - r9988;
        double r9990 = r9987 + r9988;
        double r9991 = r9989 / r9990;
        double r9992 = sqrt(r9991);
        double r9993 = atan(r9992);
        double r9994 = r9986 * r9993;
        return r9994;
}
↓
double f(double x) {
        double r9995 = 2.0;
        double r9996 = 1.0;
        double r9997 = x;
        double r9998 = r9996 - r9997;
        double r9999 = sqrt(r9998);
        double r10000 = r9996 + r9997;
        double r10001 = sqrt(r10000);
        double r10002 = r9999 / r10001;
        double r10003 = fabs(r10002);
        double r10004 = atan(r10003);
        double r10005 = r9995 * r10004;
        return r10005;
}

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-sqr-sqrt0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied rem-sqrt-square0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))