Average Error: 13.3 → 13.5
Time: 35.0s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1.0 + \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6278403065.98233:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1.0 \cdot 1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}{1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1.0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt{\mathsf{fma}\left(p, p \cdot 4.0, x \cdot x\right)}}\right)}\\ \end{array}\]
\sqrt{0.5 \cdot \left(1.0 + \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6278403065.98233:\\
\;\;\;\;\sqrt{0.5 \cdot \frac{1.0 \cdot 1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}{1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot \left(1.0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt{\mathsf{fma}\left(p, p \cdot 4.0, x \cdot x\right)}}\right)}\\

\end{array}
double f(double p, double x) {
        double r8932961 = 0.5;
        double r8932962 = 1.0;
        double r8932963 = x;
        double r8932964 = 4.0;
        double r8932965 = p;
        double r8932966 = r8932964 * r8932965;
        double r8932967 = r8932966 * r8932965;
        double r8932968 = r8932963 * r8932963;
        double r8932969 = r8932967 + r8932968;
        double r8932970 = sqrt(r8932969);
        double r8932971 = r8932963 / r8932970;
        double r8932972 = r8932962 + r8932971;
        double r8932973 = r8932961 * r8932972;
        double r8932974 = sqrt(r8932973);
        return r8932974;
}
↓
double f(double p, double x) {
        double r8932975 = x;
        double r8932976 = -6278403065.98233;
        bool r8932977 = r8932975 <= r8932976;
        double r8932978 = 0.5;
        double r8932979 = 1.0;
        double r8932980 = r8932979 * r8932979;
        double r8932981 = r8932975 * r8932975;
        double r8932982 = p;
        double r8932983 = 4.0;
        double r8932984 = r8932982 * r8932983;
        double r8932985 = r8932982 * r8932984;
        double r8932986 = r8932981 + r8932985;
        double r8932987 = sqrt(r8932986);
        double r8932988 = r8932975 / r8932987;
        double r8932989 = r8932988 * r8932988;
        double r8932990 = r8932980 - r8932989;
        double r8932991 = r8932979 - r8932988;
        double r8932992 = r8932990 / r8932991;
        double r8932993 = r8932978 * r8932992;
        double r8932994 = sqrt(r8932993);
        double r8932995 = cbrt(r8932975);
        double r8932996 = r8932995 * r8932995;
        double r8932997 = fma(r8932982, r8932984, r8932981);
        double r8932998 = sqrt(r8932997);
        double r8932999 = r8932995 / r8932998;
        double r8933000 = r8932996 * r8932999;
        double r8933001 = r8932979 + r8933000;
        double r8933002 = r8932978 * r8933001;
        double r8933003 = sqrt(r8933002);
        double r8933004 = r8932977 ? r8932994 : r8933003;
        return r8933004;
}

Error

Bits error versus p

Bits error versus x

Target

Original13.3
Target13.3
Herbie13.5
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1.0, \frac{2.0 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6278403065.98233

    1. Initial program 30.3

      \[\sqrt{0.5 \cdot \left(1.0 + \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
    2. Using strategy rm

    3. Applied flip-+30.3

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1.0 \cdot 1.0 - \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}}{1.0 - \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}}}}\]

    if -6278403065.98233 < x

    1. Initial program 8.2

      \[\sqrt{0.5 \cdot \left(1.0 + \frac{x}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity8.2

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \frac{x}{\sqrt{\color{blue}{1 \cdot \left(\left(4.0 \cdot p\right) \cdot p + x \cdot x\right)}}}\right)}\]

    4. Applied sqrt-prod8.2

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \frac{x}{\color{blue}{\sqrt{1} \cdot \sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

    5. Applied add-cube-cbrt8.4

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt{1} \cdot \sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

    6. Applied times-frac8.4

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{1}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]

    7. Simplified8.4

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{\sqrt{\left(4.0 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

    8. Simplified8.4

      \[\leadsto \sqrt{0.5 \cdot \left(1.0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\frac{\sqrt[3]{x}}{\sqrt{\mathsf{fma}\left(p, 4.0 \cdot p, x \cdot x\right)}}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6278403065.98233:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1.0 \cdot 1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}} \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}{1.0 - \frac{x}{\sqrt{x \cdot x + p \cdot \left(p \cdot 4.0\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1.0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt{\mathsf{fma}\left(p, p \cdot 4.0, x \cdot x\right)}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))