Average Error: 15.5 → 0.2
Time: 22.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000000000000004:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.1875, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \frac{0.5}{\sqrt{1}}\right)\right)}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\ \end{array}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000000000000004:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.1875, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \frac{0.5}{\sqrt{1}}\right)\right)}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\

\end{array}
double f(double x) {
        double r156568 = 1.0;
        double r156569 = 0.5;
        double r156570 = x;
        double r156571 = hypot(r156568, r156570);
        double r156572 = r156568 / r156571;
        double r156573 = r156568 + r156572;
        double r156574 = r156569 * r156573;
        double r156575 = sqrt(r156574);
        double r156576 = r156568 - r156575;
        return r156576;
}
↓
double f(double x) {
        double r156577 = 1.0;
        double r156578 = x;
        double r156579 = hypot(r156577, r156578);
        double r156580 = 1.0000000000000004;
        bool r156581 = r156579 <= r156580;
        double r156582 = 0.25;
        double r156583 = 2.0;
        double r156584 = pow(r156578, r156583);
        double r156585 = sqrt(r156577);
        double r156586 = 3.0;
        double r156587 = pow(r156585, r156586);
        double r156588 = r156584 / r156587;
        double r156589 = 0.5;
        double r156590 = 0.1875;
        double r156591 = 4.0;
        double r156592 = pow(r156578, r156591);
        double r156593 = 5.0;
        double r156594 = pow(r156585, r156593);
        double r156595 = r156592 / r156594;
        double r156596 = r156589 / r156585;
        double r156597 = fma(r156590, r156595, r156596);
        double r156598 = r156589 - r156597;
        double r156599 = fma(r156582, r156588, r156598);
        double r156600 = r156577 / r156579;
        double r156601 = r156577 + r156600;
        double r156602 = r156589 * r156601;
        double r156603 = sqrt(r156602);
        double r156604 = r156603 + r156577;
        double r156605 = r156599 / r156604;
        double r156606 = r156577 * r156577;
        double r156607 = r156601 * r156589;
        double r156608 = r156606 - r156607;
        double r156609 = log(r156608);
        double r156610 = exp(r156609);
        double r156611 = r156610 / r156604;
        double r156612 = r156581 ? r156605 : r156611;
        return r156612;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1.0 x) < 1.0000000000000004

    1. Initial program 30.4

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
    2. Using strategy rm

    3. Applied flip--30.4

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

    4. Simplified30.4

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

    5. Simplified30.4

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{\color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}}\]
    6. Using strategy rm

    7. Applied add-exp-log30.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube30.4

      \[\leadsto \frac{e^{\color{blue}{\sqrt[3]{\left(\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right) \cdot \log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

    10. Simplified30.4

      \[\leadsto \frac{e^{\sqrt[3]{\color{blue}{{\left(\log \left(\mathsf{fma}\left(1, 1, -0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)\right)}^{3}}}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

    11. Taylor expanded around 0 30.4

      \[\leadsto \frac{\color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.5\right) - \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

    12. Simplified0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.1875, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \frac{0.5}{\sqrt{1}}\right)\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]

    if 1.0000000000000004 < (hypot 1.0 x)

    1. Initial program 1.4

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
    2. Using strategy rm

    3. Applied flip--1.4

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

    4. Simplified0.5

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

    5. Simplified0.5

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{\color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}}\]
    6. Using strategy rm

    7. Applied add-exp-log0.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.0000000000000004:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.1875, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \frac{0.5}{\sqrt{1}}\right)\right)}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}}{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))