Given's Rotation SVD example, simplified

Average Error: 15.4 → 0.2

Time: 10.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.001253099710940862093408454391862960619619:\\ \;\;\;\;\frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)}\\ \mathbf{elif}\;x \le 3.762439779952672451977802037248466149322 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}\right)}^{3}}{\mathsf{fma}\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{3} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{6}\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r243957 = 1.0;
        double r243958 = 0.5;
        double r243959 = x;
        double r243960 = hypot(r243957, r243959);
        double r243961 = r243957 / r243960;
        double r243962 = r243957 + r243961;
        double r243963 = r243958 * r243962;
        double r243964 = sqrt(r243963);
        double r243965 = r243957 - r243964;
        return r243965;
}

↓

double f(double x) {
        double r243966 = x;
        double r243967 = -0.001253099710940862;
        bool r243968 = r243966 <= r243967;
        double r243969 = 1.0;
        double r243970 = 3.0;
        double r243971 = pow(r243969, r243970);
        double r243972 = 0.5;
        double r243973 = hypot(r243969, r243966);
        double r243974 = r243969 / r243973;
        double r243975 = r243969 + r243974;
        double r243976 = r243972 * r243975;
        double r243977 = 1.5;
        double r243978 = pow(r243976, r243977);
        double r243979 = r243971 - r243978;
        double r243980 = pow(r243974, r243970);
        double r243981 = r243971 + r243980;
        double r243982 = r243972 * r243981;
        double r243983 = sqrt(r243982);
        double r243984 = r243969 * r243969;
        double r243985 = r243974 * r243974;
        double r243986 = r243969 * r243974;
        double r243987 = r243985 - r243986;
        double r243988 = r243984 + r243987;
        double r243989 = sqrt(r243988);
        double r243990 = r243983 / r243989;
        double r243991 = r243969 * r243990;
        double r243992 = fma(r243972, r243975, r243991);
        double r243993 = fma(r243969, r243969, r243992);
        double r243994 = r243979 / r243993;
        double r243995 = 0.00037624397799526725;
        bool r243996 = r243966 <= r243995;
        double r243997 = 0.25;
        double r243998 = 2.0;
        double r243999 = pow(r243966, r243998);
        double r244000 = sqrt(r243969);
        double r244001 = pow(r244000, r243970);
        double r244002 = r243999 / r244001;
        double r244003 = 1.0;
        double r244004 = r244003 / r244000;
        double r244005 = 0.1875;
        double r244006 = 4.0;
        double r244007 = pow(r243966, r244006);
        double r244008 = 5.0;
        double r244009 = pow(r244000, r244008);
        double r244010 = r244007 / r244009;
        double r244011 = r244005 * r244010;
        double r244012 = fma(r243972, r244004, r244011);
        double r244013 = r243972 - r244012;
        double r244014 = fma(r243997, r244002, r244013);
        double r244015 = sqrt(r243976);
        double r244016 = r243969 + r244015;
        double r244017 = r244014 / r244016;
        double r244018 = pow(r243971, r243970);
        double r244019 = pow(r243978, r243970);
        double r244020 = r244018 - r244019;
        double r244021 = r243971 + r243978;
        double r244022 = 6.0;
        double r244023 = pow(r243969, r244022);
        double r244024 = fma(r243978, r244021, r244023);
        double r244025 = r244020 / r244024;
        double r244026 = r243969 * r244015;
        double r244027 = fma(r243972, r243975, r244026);
        double r244028 = fma(r243969, r243969, r244027);
        double r244029 = r244025 / r244028;
        double r244030 = r243996 ? r244017 : r244029;
        double r244031 = r243968 ? r243994 : r244030;
        return r244031;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.001253099710940862

   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 flip3-- 1.6

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

   4. Simplified 1.1

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

   6. Applied pow1/2 1.1

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

   7. Applied pow-pow 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\color{blue}{\frac{3}{2}}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
   9. Using strategy rm

   10. Applied flip3-+ 0.1

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

   11. Applied associate-*r/ 0.1

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

   12. Applied sqrt-div 0.1

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

   if -0.001253099710940862 < x < 0.00037624397799526725

   1. Initial program 29.9

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

   3. Applied flip-- 29.9

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

   4. Simplified 29.9

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

   5. Taylor expanded around 0 29.9

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

   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 0.00037624397799526725 < x

   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 flip3-- 1.6

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

   4. Simplified 1.0

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

   6. Applied pow1/2 1.0

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

   7. Applied pow-pow 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\color{blue}{\frac{3}{2}}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
   9. Using strategy rm

   10. Applied flip3-- 1.0

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

   11. Simplified 0.1

       \[\leadsto \frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{3} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{6}\right)}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001253099710940862093408454391862960619619:\\ \;\;\;\;\frac{{1}^{3} - {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)}\\ \mathbf{elif}\;x \le 3.762439779952672451977802037248466149322 \cdot 10^{-4}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left({1}^{3}\right)}^{3} - {\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}\right)}^{3}}{\mathsf{fma}\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{3} + {\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{\frac{3}{2}}, {1}^{6}\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))