Average Error: 0.0 → 0.0
Time: 25.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r53239 = 2.0;
        double r53240 = 1.0;
        double r53241 = x;
        double r53242 = r53240 - r53241;
        double r53243 = r53240 + r53241;
        double r53244 = r53242 / r53243;
        double r53245 = sqrt(r53244);
        double r53246 = atan(r53245);
        double r53247 = r53239 * r53246;
        return r53247;
}
↓
double f(double x) {
        double r53248 = 2.0;
        double r53249 = 1.0;
        double r53250 = x;
        double r53251 = r53249 - r53250;
        double r53252 = sqrt(r53251);
        double r53253 = r53249 + r53250;
        double r53254 = sqrt(r53253);
        double r53255 = r53252 / r53254;
        double r53256 = r53255 * r53255;
        double r53257 = sqrt(r53256);
        double r53258 = atan(r53257);
        double r53259 = r53248 * r53258;
        return r53259;
}

Error

Bits error versus x

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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. Final simplification0.0

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

Reproduce

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