Given's Rotation SVD example, simplified

Average Error: 15.4 → 0.2

Time: 16.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0023638476354078493:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}, 1 \cdot 1\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 0.001099911291066277:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r309674 = 1.0;
        double r309675 = 0.5;
        double r309676 = x;
        double r309677 = hypot(r309674, r309676);
        double r309678 = r309674 / r309677;
        double r309679 = r309674 + r309678;
        double r309680 = r309675 * r309679;
        double r309681 = sqrt(r309680);
        double r309682 = r309674 - r309681;
        return r309682;
}

↓

double f(double x) {
        double r309683 = x;
        double r309684 = -0.0023638476354078493;
        bool r309685 = r309683 <= r309684;
        double r309686 = 0.5;
        double r309687 = -r309686;
        double r309688 = 1.0;
        double r309689 = hypot(r309688, r309683);
        double r309690 = r309688 / r309689;
        double r309691 = 3.0;
        double r309692 = pow(r309690, r309691);
        double r309693 = cbrt(r309692);
        double r309694 = r309688 + r309693;
        double r309695 = r309688 * r309688;
        double r309696 = fma(r309687, r309694, r309695);
        double r309697 = r309688 + r309690;
        double r309698 = r309686 * r309697;
        double r309699 = sqrt(r309698);
        double r309700 = r309688 + r309699;
        double r309701 = r309696 / r309700;
        double r309702 = 0.0010999112910662773;
        bool r309703 = r309683 <= r309702;
        double r309704 = 0.25;
        double r309705 = 2.0;
        double r309706 = pow(r309683, r309705);
        double r309707 = sqrt(r309688);
        double r309708 = pow(r309707, r309691);
        double r309709 = r309706 / r309708;
        double r309710 = 1.0;
        double r309711 = r309710 / r309707;
        double r309712 = 0.1875;
        double r309713 = 4.0;
        double r309714 = pow(r309683, r309713);
        double r309715 = 5.0;
        double r309716 = pow(r309707, r309715);
        double r309717 = r309714 / r309716;
        double r309718 = r309712 * r309717;
        double r309719 = fma(r309686, r309711, r309718);
        double r309720 = r309686 - r309719;
        double r309721 = fma(r309704, r309709, r309720);
        double r309722 = r309721 / r309700;
        double r309723 = fma(r309687, r309697, r309695);
        double r309724 = log1p(r309700);
        double r309725 = log1p(r309724);
        double r309726 = expm1(r309725);
        double r309727 = expm1(r309726);
        double r309728 = r309723 / r309727;
        double r309729 = r309703 ? r309722 : r309728;
        double r309730 = r309685 ? r309701 : r309729;
        return r309730;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.0023638476354078493

   1. Initial program 1.0

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

   3. Applied flip-- 1.0

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

   4. Simplified 0.1

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

   6. Applied add-cbrt-cube 0.1

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

   7. Applied add-cbrt-cube 0.1

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

   8. Applied cbrt-undiv 0.1

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

   9. Simplified 0.1

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

   if -0.0023638476354078493 < x < 0.0010999112910662773

   1. Initial program 30.1

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

   3. Applied flip-- 30.1

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

   4. Simplified 30.1

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

   5. Taylor expanded around 0 30.1

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

   6. Simplified 0.3

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

   if 0.0010999112910662773 < x

   1. Initial program 1.1

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

   3. Applied flip-- 1.1

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

   4. Simplified 0.1

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

   6. Applied expm1-log1p-u 0.1

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}}\]
   7. Using strategy rm

   8. Applied expm1-log1p-u 0.1

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0023638476354078493:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}, 1 \cdot 1\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 0.001099911291066277:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))