Average Error: 0.0 → 0.0
Time: 5.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{1}^{3} + {x}^{3}} \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{1}^{3} + {x}^{3}} \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\right)
double f(double x) {
        double r16566 = 2.0;
        double r16567 = 1.0;
        double r16568 = x;
        double r16569 = r16567 - r16568;
        double r16570 = r16567 + r16568;
        double r16571 = r16569 / r16570;
        double r16572 = sqrt(r16571);
        double r16573 = atan(r16572);
        double r16574 = r16566 * r16573;
        return r16574;
}
↓
double f(double x) {
        double r16575 = 2.0;
        double r16576 = 1.0;
        double r16577 = x;
        double r16578 = r16576 - r16577;
        double r16579 = 3.0;
        double r16580 = pow(r16576, r16579);
        double r16581 = pow(r16577, r16579);
        double r16582 = r16580 + r16581;
        double r16583 = r16578 / r16582;
        double r16584 = r16576 * r16576;
        double r16585 = r16577 * r16577;
        double r16586 = r16576 * r16577;
        double r16587 = r16585 - r16586;
        double r16588 = r16584 + r16587;
        double r16589 = r16583 * r16588;
        double r16590 = sqrt(r16589);
        double r16591 = atan(r16590);
        double r16592 = r16575 * r16591;
        return r16592;
}

Error

Bits error versus x

Try it out

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. 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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))