Average Error: 0.0 → 0.0
Time: 7.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \frac{1 - x}{{1}^{3} + {x}^{3}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \frac{1 - x}{{1}^{3} + {x}^{3}}}\right) \cdot 2
double f(double x) {
        double r15762 = 2.0;
        double r15763 = 1.0;
        double r15764 = x;
        double r15765 = r15763 - r15764;
        double r15766 = r15763 + r15764;
        double r15767 = r15765 / r15766;
        double r15768 = sqrt(r15767);
        double r15769 = atan(r15768);
        double r15770 = r15762 * r15769;
        return r15770;
}
↓
double f(double x) {
        double r15771 = 1.0;
        double r15772 = r15771 * r15771;
        double r15773 = x;
        double r15774 = r15773 * r15773;
        double r15775 = r15771 * r15773;
        double r15776 = r15774 - r15775;
        double r15777 = r15772 + r15776;
        double r15778 = r15771 - r15773;
        double r15779 = 3.0;
        double r15780 = pow(r15771, r15779);
        double r15781 = pow(r15773, r15779);
        double r15782 = r15780 + r15781;
        double r15783 = r15778 / r15782;
        double r15784 = r15777 * r15783;
        double r15785 = sqrt(r15784);
        double r15786 = atan(r15785);
        double r15787 = 2.0;
        double r15788 = r15786 * r15787;
        return r15788;
}

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. Simplified0.0

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))