Average Error: 0.0 → 0.0
Time: 4.3s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}} \cdot \frac{1 - x}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}} \cdot \frac{1 - x}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r16410 = 2.0;
        double r16411 = 1.0;
        double r16412 = x;
        double r16413 = r16411 - r16412;
        double r16414 = r16411 + r16412;
        double r16415 = r16413 / r16414;
        double r16416 = sqrt(r16415);
        double r16417 = atan(r16416);
        double r16418 = r16410 * r16417;
        return r16418;
}
↓
double f(double x) {
        double r16419 = 2.0;
        double r16420 = 1.0;
        double r16421 = 1.0;
        double r16422 = x;
        double r16423 = r16421 + r16422;
        double r16424 = sqrt(r16423);
        double r16425 = r16420 / r16424;
        double r16426 = r16421 - r16422;
        double r16427 = r16426 / r16424;
        double r16428 = r16425 * r16427;
        double r16429 = sqrt(r16428);
        double r16430 = atan(r16429);
        double r16431 = r16419 * r16430;
        return r16431;
}

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 *-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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))