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+---
+layout: post
+title: 快速平方根倒数
+date: 2024-04-23 13:00:00 +0800
+categories: algorithm
+tags: bit-hack
+published: true
+---
+
+* content
+{:toc}
+
+## 雷神之锤3源码
+
+```C
+float Q_rsqrt( float number )
+{
+	long i;
+	float x2, y;
+	const float threehalfs = 1.5F;
+
+	x2 = number * 0.5F;
+	y  = number;
+	i  = * ( long * ) &y;                       // evil floating point bit level hacking
+	i  = 0x5f3759df - ( i >> 1 );               // what the fuck?
+	y  = * ( float * ) &i;
+	y  = y * ( threehalfs - ( x2 * y * y ) );   // 1st iteration
+//  y  = y * ( threehalfs - ( x2 * y * y ) );   // 2nd iteration, this can be removed
+
+	return y;
+}
+```
+
+## 牛顿迭代
+
+
+
+## IEEE 754
+
+科学计数法
+
+normalised numbers
+denormalised numbers
+NaN
+0 & -0
+
+## 推导
+
+x 为 float，令 y 为 x 的平方根倒数：
+$$
+\begin{aligned}
+& \quad\,y=\frac{1}{\sqrt{x}}={x^{-s\frac{1}{2}}}\\
+&\quad\,y=\frac{1}{\sqrt{x}}={x^{-s\frac{1}{2}}}\\
+&\Rightarrow\log_2 (y)=-\frac{1}{2}\log_2 (x)\\
+\Rightarrow\log_2 (y)=-\frac{1}{2}\log_2 (x)\\
+$$
+
+代入 x 和 y float 的 long representation
+
+$$
+&\rightarrow \log_2 (2^(E_y-127)*(1.0+\frac{M_y}{2^23}))=-\frac{1}{2}\log_2 (2^(E_x-127)*(1.0+\frac{M_x}{2^23}))\\
+&⇒=\sum_{i=1}^{n-k}{i^2} + (2n-k+1)\sum_{i=n-k+1}^{n}{i} - \sum_{i=n-k+1}^{n}{i^2}\\
+&=\sum_{i=1}^{n-k}{i^2} + (2n-k+1)(\sum_{i=1}^{n}{i}-\sum_{i=1}^{n-k}{i}) - (\sum_{i=1}^{n}{i^2}-\sum_{i=1}^{n-k}{i^2})\\
+&=2\sum_{i=1}^{n-k}{i^2} - \sum_{i=1}^{n}{i^2}  + (2n-k+1)(\sum_{i=1}^{n}{i}-\sum_{i=1}^{n-k}{i})\\
+&=2\frac{(n-k)[(n-k)+1][2(n-k)+1]}{6} - \frac{n(n+1)(2n+1)}{6} + (2n-k+1)(\frac{n(n+1)}{2}-\frac{(n-k)(n-k+1)}{2})\\
+&=\frac{2n^3+3n^2+n-k^3+k}{6}
+\end{aligned}
+$$
+
+$\log_2 10$
+
+$$\lg_2 10$$
+
+$$\lg 10$$
+
+$$lg 10$$
+
+$$\ln 10$$
+
+$$ln 10$$
+
+$$\log_2 10$$
+
+$$log_2 10$$
+
+$\sqrt{3x-1}+(1+x)^2$
+
+$$\sqrt3$$
+
+$$\sqrt{3}$$
+
+$$\sqrt{3x-1}$$
+
+$$\sqrt{3x-1}+(1+x)^2$$
+
+$10^{10}$
+
+$\frac{4n^3-6n^2-10n-2k^3-3k^2+8k+12kn+c}{12}$
+
+$$\frac{4n^3-6n^2-10n-2k^3-3k^2+8k+12kn+c}{12}$$
+
+$$\frac{4}{12}$$
+
+$$\tfrac{4}{12}$$
+
+$$\dfrac{4}{12}$$
+
+$$\cfrac{4}{12}$$
+
+## 总结
+
+现在已经有了平方根运算器，已经不需要这么写了。
+
+<!-- https://www.youtube.com/watch?v=p8u_k2LIZyo -->
+<!-- https://www.zhihu.com/question/26287650 -->
+<!-- https://en.wikipedia.org/wiki/Fast_inverse_square_root -->
+<!-- https://zh.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9%E5%80%92%E6%95%B0%E9%80%9F%E7%AE%97%E6%B3%95 -->
+<!-- https://zhuanlan.zhihu.com/p/400064205 -->
+<!-- https://brilliant.org/wiki/newton-raphson-method/ -->
\ No newline at end of file