Average Error: 0.0 → 0.0
Time: 23.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r49009 = 2.0;
        double r49010 = 1.0;
        double r49011 = x;
        double r49012 = r49010 - r49011;
        double r49013 = r49010 + r49011;
        double r49014 = r49012 / r49013;
        double r49015 = sqrt(r49014);
        double r49016 = atan(r49015);
        double r49017 = r49009 * r49016;
        return r49017;
}
↓
double f(double x) {
        double r49018 = 2.0;
        double r49019 = 1.0;
        double r49020 = x;
        double r49021 = r49019 - r49020;
        double r49022 = r49019 + r49020;
        double r49023 = sqrt(r49022);
        double r49024 = r49021 / r49023;
        double r49025 = r49024 / r49023;
        double r49026 = sqrt(r49025);
        double r49027 = atan(r49026);
        double r49028 = r49018 * r49027;
        return r49028;
}

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 associate-/r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))