Average Error: 0.0 → 0.0
Time: 2.5s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{t_0}}{t_0}}\right) \end{array} \]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\begin{array}{l}
t_0 := \sqrt{1 + x}\\
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{t_0}}{t_0}}\right)
\end{array}

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (* 2.0 (atan (sqrt (/ (/ (- 1.0 x) t_0) t_0))))))
double code(double x) {
	return 2.0 * atan(sqrt((1.0 - x) / (1.0 + x)));
}
↓
double code(double x) {
	double t_0 = sqrt(1.0 + x);
	return 2.0 * atan(sqrt(((1.0 - x) / t_0) / t_0));
}

Error

Bits error versus x

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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. Applied add-sqr-sqrt_binary640.0

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

  3. Applied associate-/r*_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2021313 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))