Average Error: 0.0 → 0.0
Time: 17.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-\sqrt[3]{x}}{1 + x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right) + \mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-\sqrt[3]{x}}{1 + x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right) + \mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right)}\right)
double f(double x) {
        double r292172 = 2.0;
        double r292173 = 1.0;
        double r292174 = x;
        double r292175 = r292173 - r292174;
        double r292176 = r292173 + r292174;
        double r292177 = r292175 / r292176;
        double r292178 = sqrt(r292177);
        double r292179 = atan(r292178);
        double r292180 = r292172 * r292179;
        return r292180;
}
↓
double f(double x) {
        double r292181 = 2.0;
        double r292182 = x;
        double r292183 = cbrt(r292182);
        double r292184 = -r292183;
        double r292185 = 1.0;
        double r292186 = r292185 + r292182;
        double r292187 = r292184 / r292186;
        double r292188 = r292183 * r292183;
        double r292189 = r292183 / r292186;
        double r292190 = r292188 * r292189;
        double r292191 = fma(r292187, r292188, r292190);
        double r292192 = r292182 * r292182;
        double r292193 = r292185 - r292192;
        double r292194 = r292185 / r292193;
        double r292195 = r292185 - r292182;
        double r292196 = -r292190;
        double r292197 = fma(r292194, r292195, r292196);
        double r292198 = r292191 + r292197;
        double r292199 = sqrt(r292198);
        double r292200 = atan(r292199);
        double r292201 = r292181 * r292200;
        return r292201;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied div-sub0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied *-un-lft-identity0.0

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

  7. Applied distribute-lft-out0.0

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

  8. Applied add-cube-cbrt0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{1 + x} - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot \left(1 + x\right)}}\right)\]

  9. Applied times-frac0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{1 + x} - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{1 + x}}}\right)\]

  10. Applied flip-+0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{1 + x}}\right)\]

  11. Applied associate-/r/0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right)} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{1 + x}}\right)\]

  12. Applied prod-diff0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{1 \cdot 1 - x \cdot x}, 1 - x, -\frac{\sqrt[3]{x}}{1 + x} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{x}}{1 + x}, \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{1 + x} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}}\right)\]

  13. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-\sqrt[3]{x}}{1 + x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right) + \mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{1 + x}\right)}\right)\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))