Given's Rotation SVD example, simplified

Average Error: 15.3 → 0.0

Time: 18.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.010054607689223198:\\ \;\;\;\;\frac{\frac{e^{\log \left(\sqrt[3]{\left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right) \cdot \left(\left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right) \cdot \left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right)\right)}\right)}}{\mathsf{fma}\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}, 1 + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)}}{\sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} + 1}\\ \mathbf{elif}\;x \le 0.010372877460017476:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{-11}{128}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{69}{1024}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)}{\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) + \frac{1}{2}} + 1}\\ \end{array}\]

C

double f(double x) {
        double r2448746 = 1.0;
        double r2448747 = 0.5;
        double r2448748 = x;
        double r2448749 = hypot(r2448746, r2448748);
        double r2448750 = r2448746 / r2448749;
        double r2448751 = r2448746 + r2448750;
        double r2448752 = r2448747 * r2448751;
        double r2448753 = sqrt(r2448752);
        double r2448754 = r2448746 - r2448753;
        return r2448754;
}

↓

double f(double x) {
        double r2448755 = x;
        double r2448756 = -0.010054607689223198;
        bool r2448757 = r2448755 <= r2448756;
        double r2448758 = 1.0;
        double r2448759 = 0.5;
        double r2448760 = hypot(r2448758, r2448755);
        double r2448761 = r2448759 / r2448760;
        double r2448762 = r2448761 + r2448759;
        double r2448763 = r2448762 * r2448762;
        double r2448764 = r2448762 * r2448763;
        double r2448765 = r2448758 - r2448764;
        double r2448766 = r2448765 * r2448765;
        double r2448767 = r2448765 * r2448766;
        double r2448768 = cbrt(r2448767);
        double r2448769 = log(r2448768);
        double r2448770 = exp(r2448769);
        double r2448771 = r2448758 + r2448762;
        double r2448772 = fma(r2448762, r2448762, r2448771);
        double r2448773 = r2448770 / r2448772;
        double r2448774 = sqrt(r2448762);
        double r2448775 = r2448774 + r2448758;
        double r2448776 = r2448773 / r2448775;
        double r2448777 = 0.010372877460017476;
        bool r2448778 = r2448755 <= r2448777;
        double r2448779 = r2448755 * r2448755;
        double r2448780 = r2448779 * r2448779;
        double r2448781 = -0.0859375;
        double r2448782 = 0.125;
        double r2448783 = r2448779 * r2448780;
        double r2448784 = 0.0673828125;
        double r2448785 = r2448783 * r2448784;
        double r2448786 = fma(r2448779, r2448782, r2448785);
        double r2448787 = fma(r2448780, r2448781, r2448786);
        double r2448788 = r2448758 - r2448762;
        double r2448789 = exp(r2448761);
        double r2448790 = log(r2448789);
        double r2448791 = r2448790 + r2448759;
        double r2448792 = sqrt(r2448791);
        double r2448793 = r2448792 + r2448758;
        double r2448794 = r2448788 / r2448793;
        double r2448795 = r2448778 ? r2448787 : r2448794;
        double r2448796 = r2448757 ? r2448776 : r2448795;
        return r2448796;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.010054607689223198

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   8. Applied flip3-- 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

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

   12. Applied add-exp-log 0.1

       \[\leadsto \frac{\frac{\color{blue}{e^{\log \left(1 - \left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + 1\right)}}{\sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} + 1}\]
   13. Using strategy rm

   14. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{\frac{e^{\log \color{blue}{\left(\sqrt[3]{\left(\left(1 - \left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(1 - \left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(1 - \left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)}\right)}}}{\mathsf{fma}\left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}, \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + 1\right)}}{\sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} + 1}\]

   if -0.010054607689223198 < x < 0.010372877460017476

   1. Initial program 29.9

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

   2. Simplified 29.9

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

   3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{69}{1024} \cdot {x}^{6}\right) - \frac{11}{128} \cdot {x}^{4}}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{-11}{128}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \frac{69}{1024} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\]

   if 0.010372877460017476 < x

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   8. Applied add-log-exp 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.010054607689223198:\\ \;\;\;\;\frac{\frac{e^{\log \left(\sqrt[3]{\left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right) \cdot \left(\left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right) \cdot \left(1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)\right)\right)}\right)}}{\mathsf{fma}\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}, \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}, 1 + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)\right)}}{\sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} + 1}\\ \mathbf{elif}\;x \le 0.010372877460017476:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{-11}{128}, \mathsf{fma}\left(x \cdot x, \frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{69}{1024}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right)}{\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) + \frac{1}{2}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))