Average Error: 0.0 → 0.0
Time: 2.1s
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 r5270 = 2.0;
        double r5271 = 1.0;
        double r5272 = x;
        double r5273 = r5271 - r5272;
        double r5274 = r5271 + r5272;
        double r5275 = r5273 / r5274;
        double r5276 = sqrt(r5275);
        double r5277 = atan(r5276);
        double r5278 = r5270 * r5277;
        return r5278;
}
↓
double f(double x) {
        double r5279 = 2.0;
        double r5280 = 1.0;
        double r5281 = x;
        double r5282 = r5280 - r5281;
        double r5283 = sqrt(r5282);
        double r5284 = r5280 + r5281;
        double r5285 = sqrt(r5284);
        double r5286 = r5283 / r5285;
        double r5287 = fabs(r5286);
        double r5288 = atan(r5287);
        double r5289 = r5279 * r5288;
        return r5289;
}

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. 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 2020003 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))