Average Error: 0.0 → 0.0
Time: 26.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 - x \cdot x}} \cdot \sqrt{1 - x}}{\sqrt{1 + x}}}\right)\]
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 \cdot x}} \cdot \sqrt{1 - x}}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r1150194 = 2.0;
        double r1150195 = 1.0;
        double r1150196 = x;
        double r1150197 = r1150195 - r1150196;
        double r1150198 = r1150195 + r1150196;
        double r1150199 = r1150197 / r1150198;
        double r1150200 = sqrt(r1150199);
        double r1150201 = atan(r1150200);
        double r1150202 = r1150194 * r1150201;
        return r1150202;
}
↓
double f(double x) {
        double r1150203 = 2.0;
        double r1150204 = 1.0;
        double r1150205 = x;
        double r1150206 = r1150204 - r1150205;
        double r1150207 = r1150205 * r1150205;
        double r1150208 = r1150204 - r1150207;
        double r1150209 = sqrt(r1150208);
        double r1150210 = r1150206 / r1150209;
        double r1150211 = sqrt(r1150206);
        double r1150212 = r1150210 * r1150211;
        double r1150213 = r1150204 + r1150205;
        double r1150214 = sqrt(r1150213);
        double r1150215 = r1150212 / r1150214;
        double r1150216 = sqrt(r1150215);
        double r1150217 = atan(r1150216);
        double r1150218 = r1150203 * r1150217;
        return r1150218;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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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. Using strategy rm

  6. Applied flip-+0.0

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

  7. Applied sqrt-div0.0

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

  8. Applied associate-/r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))