Average Error: 16.0 → 0.1
Time: 24.9s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000002351709365:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-3}{16}\right)\right)}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \left(\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)} + 1}\\ \end{array}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000002351709365:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-3}{16}\right)\right)}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \left(\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)} + 1}\\

\end{array}
double f(double x) {
        double r3723353 = 1.0;
        double r3723354 = 0.5;
        double r3723355 = x;
        double r3723356 = hypot(r3723353, r3723355);
        double r3723357 = r3723353 / r3723356;
        double r3723358 = r3723353 + r3723357;
        double r3723359 = r3723354 * r3723358;
        double r3723360 = sqrt(r3723359);
        double r3723361 = r3723353 - r3723360;
        return r3723361;
}
↓
double f(double x) {
        double r3723362 = 1.0;
        double r3723363 = x;
        double r3723364 = hypot(r3723362, r3723363);
        double r3723365 = 1.0000002351709365;
        bool r3723366 = r3723364 <= r3723365;
        double r3723367 = 0.25;
        double r3723368 = r3723363 * r3723363;
        double r3723369 = r3723368 * r3723368;
        double r3723370 = r3723369 * r3723368;
        double r3723371 = 0.15625;
        double r3723372 = -0.1875;
        double r3723373 = r3723369 * r3723372;
        double r3723374 = fma(r3723370, r3723371, r3723373);
        double r3723375 = fma(r3723367, r3723368, r3723374);
        double r3723376 = 0.5;
        double r3723377 = r3723376 / r3723364;
        double r3723378 = r3723376 + r3723377;
        double r3723379 = sqrt(r3723378);
        double r3723380 = r3723379 + r3723362;
        double r3723381 = r3723375 / r3723380;
        double r3723382 = r3723376 - r3723377;
        double r3723383 = cbrt(r3723378);
        double r3723384 = r3723383 * r3723383;
        double r3723385 = r3723383 * r3723384;
        double r3723386 = sqrt(r3723385);
        double r3723387 = r3723386 + r3723362;
        double r3723388 = r3723382 / r3723387;
        double r3723389 = r3723366 ? r3723381 : r3723388;
        return r3723389;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1 x) < 1.0000002351709365

    1. Initial program 30.9

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

    2. Simplified30.9

      \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
    3. Using strategy rm

    4. Applied flip--30.9

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

    5. Simplified30.8

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

    6. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \frac{5}{32} \cdot {x}^{6}\right) - \frac{3}{16} \cdot {x}^{4}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \frac{-3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

    if 1.0000002351709365 < (hypot 1 x)

    1. Initial program 1.1

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

    2. Simplified1.1

      \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
    3. Using strategy rm

    4. Applied flip--1.1

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

    5. Simplified0.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000002351709365:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \frac{5}{32}, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-3}{16}\right)\right)}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \left(\sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right)} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))