Average Error: 0.0 → 0.0
Time: 13.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{1}{{1}^{3} + {x}^{3}}, \mathsf{fma}\left(1, 1, x \cdot \left(x - 1\right)\right), -\frac{\sqrt[3]{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{1 + x}}\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{1 + x}} \cdot \left(\left(-\frac{\sqrt[3]{x}}{\sqrt{1 + x}}\right) + \frac{\sqrt[3]{x}}{\sqrt{1 + x}}\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{1}{{1}^{3} + {x}^{3}}, \mathsf{fma}\left(1, 1, x \cdot \left(x - 1\right)\right), -\frac{\sqrt[3]{x}}{\sqrt{1 + x}} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{1 + x}}\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{1 + x}} \cdot \left(\left(-\frac{\sqrt[3]{x}}{\sqrt{1 + x}}\right) + \frac{\sqrt[3]{x}}{\sqrt{1 + x}}\right)}\right)
double f(double x) {
        double r25968 = 2.0;
        double r25969 = 1.0;
        double r25970 = x;
        double r25971 = r25969 - r25970;
        double r25972 = r25969 + r25970;
        double r25973 = r25971 / r25972;
        double r25974 = sqrt(r25973);
        double r25975 = atan(r25974);
        double r25976 = r25968 * r25975;
        return r25976;
}
↓
double f(double x) {
        double r25977 = 2.0;
        double r25978 = 1.0;
        double r25979 = 3.0;
        double r25980 = pow(r25978, r25979);
        double r25981 = x;
        double r25982 = pow(r25981, r25979);
        double r25983 = r25980 + r25982;
        double r25984 = r25978 / r25983;
        double r25985 = r25981 - r25978;
        double r25986 = r25981 * r25985;
        double r25987 = fma(r25978, r25978, r25986);
        double r25988 = cbrt(r25981);
        double r25989 = r25978 + r25981;
        double r25990 = sqrt(r25989);
        double r25991 = r25988 / r25990;
        double r25992 = r25988 * r25988;
        double r25993 = r25992 / r25990;
        double r25994 = r25991 * r25993;
        double r25995 = -r25994;
        double r25996 = fma(r25984, r25987, r25995);
        double r25997 = -r25991;
        double r25998 = r25997 + r25991;
        double r25999 = r25993 * r25998;
        double r26000 = r25996 + r25999;
        double r26001 = sqrt(r26000);
        double r26002 = atan(r26001);
        double r26003 = r25977 * r26002;
        return r26003;
}

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 add-sqr-sqrt0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied times-frac0.0

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

  8. Applied flip3-+0.0

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

  9. Applied associate-/r/0.0

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

  10. Applied prod-diff0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))