Average Error: 0.0 → 0.0
Time: 28.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\left(1 + \left(x \cdot x - x\right)\right) \cdot \frac{1 - x}{1 + {x}^{3}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\left(1 + \left(x \cdot x - x\right)\right) \cdot \frac{1 - x}{1 + {x}^{3}}}\right)
double f(double x) {
        double r1154572 = 2.0;
        double r1154573 = 1.0;
        double r1154574 = x;
        double r1154575 = r1154573 - r1154574;
        double r1154576 = r1154573 + r1154574;
        double r1154577 = r1154575 / r1154576;
        double r1154578 = sqrt(r1154577);
        double r1154579 = atan(r1154578);
        double r1154580 = r1154572 * r1154579;
        return r1154580;
}
↓
double f(double x) {
        double r1154581 = 2.0;
        double r1154582 = 1.0;
        double r1154583 = x;
        double r1154584 = r1154583 * r1154583;
        double r1154585 = r1154584 - r1154583;
        double r1154586 = r1154582 + r1154585;
        double r1154587 = r1154582 - r1154583;
        double r1154588 = 3.0;
        double r1154589 = pow(r1154583, r1154588);
        double r1154590 = r1154582 + r1154589;
        double r1154591 = r1154587 / r1154590;
        double r1154592 = r1154586 * r1154591;
        double r1154593 = sqrt(r1154592);
        double r1154594 = atan(r1154593);
        double r1154595 = r1154581 * r1154594;
        return r1154595;
}

Error

Bits error versus x

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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 flip3-+0.0

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

  4. Applied associate-/r/0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))