Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1 - x}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1 - x}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r11377 = 2.0;
        double r11378 = 1.0;
        double r11379 = x;
        double r11380 = r11378 - r11379;
        double r11381 = r11378 + r11379;
        double r11382 = r11380 / r11381;
        double r11383 = sqrt(r11382);
        double r11384 = atan(r11383);
        double r11385 = r11377 * r11384;
        return r11385;
}
↓
double f(double x) {
        double r11386 = 2.0;
        double r11387 = 1.0;
        double r11388 = 1.0;
        double r11389 = x;
        double r11390 = r11388 + r11389;
        double r11391 = sqrt(r11390);
        double r11392 = r11387 / r11391;
        double r11393 = sqrt(r11392);
        double r11394 = r11388 - r11389;
        double r11395 = r11394 / r11391;
        double r11396 = sqrt(r11395);
        double r11397 = r11393 * r11396;
        double r11398 = atan(r11397);
        double r11399 = r11386 * r11398;
        return r11399;
}

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 *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Applied sqrt-prod0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))