Given's Rotation SVD example, simplified

Average Error: 15.6 → 0.2

Time: 19.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.617771452625160211354071737588355972548 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 \cdot 1 - \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{\left(1 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 0.5} \cdot 1}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 5.078281910445067021120379990861692931503 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{3}}} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.0078125, \mathsf{fma}\left(0.125, \frac{\sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \left(x \cdot x\right)}{1}, 1 - \mathsf{fma}\left(\frac{0.09375}{1} \cdot \frac{{x}^{4}}{1}, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \sqrt{1} \cdot \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \mathsf{fma}\left(1, 1, \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \end{array}\]

C

double f(double x) {
        double r205911 = 1.0;
        double r205912 = 0.5;
        double r205913 = x;
        double r205914 = hypot(r205911, r205913);
        double r205915 = r205911 / r205914;
        double r205916 = r205911 + r205915;
        double r205917 = r205912 * r205916;
        double r205918 = sqrt(r205917);
        double r205919 = r205911 - r205918;
        return r205919;
}

↓

double f(double x) {
        double r205920 = x;
        double r205921 = -0.000361777145262516;
        bool r205922 = r205920 <= r205921;
        double r205923 = 1.0;
        double r205924 = r205923 * r205923;
        double r205925 = 0.5;
        double r205926 = hypot(r205923, r205920);
        double r205927 = r205925 / r205926;
        double r205928 = r205925 + r205927;
        double r205929 = r205928 * r205928;
        double r205930 = r205924 - r205929;
        double r205931 = log(r205930);
        double r205932 = exp(r205931);
        double r205933 = r205923 + r205927;
        double r205934 = r205933 + r205925;
        double r205935 = r205932 / r205934;
        double r205936 = r205935 * r205923;
        double r205937 = r205923 * r205928;
        double r205938 = sqrt(r205937);
        double r205939 = r205923 + r205938;
        double r205940 = r205936 / r205939;
        double r205941 = 0.0005078281910445067;
        bool r205942 = r205920 <= r205941;
        double r205943 = 1.0;
        double r205944 = sqrt(r205923);
        double r205945 = r205925 / r205944;
        double r205946 = r205925 + r205945;
        double r205947 = 3.0;
        double r205948 = pow(r205946, r205947);
        double r205949 = r205943 / r205948;
        double r205950 = sqrt(r205949);
        double r205951 = 4.0;
        double r205952 = pow(r205920, r205951);
        double r205953 = 5.0;
        double r205954 = pow(r205944, r205953);
        double r205955 = r205952 / r205954;
        double r205956 = r205950 * r205955;
        double r205957 = 0.0078125;
        double r205958 = 0.125;
        double r205959 = r205943 / r205946;
        double r205960 = sqrt(r205959);
        double r205961 = r205920 * r205920;
        double r205962 = r205960 * r205961;
        double r205963 = r205962 / r205923;
        double r205964 = 0.09375;
        double r205965 = r205964 / r205923;
        double r205966 = r205952 / r205923;
        double r205967 = r205965 * r205966;
        double r205968 = sqrt(r205946);
        double r205969 = r205944 * r205968;
        double r205970 = fma(r205967, r205960, r205969);
        double r205971 = r205923 - r205970;
        double r205972 = fma(r205958, r205963, r205971);
        double r205973 = fma(r205956, r205957, r205972);
        double r205974 = r205924 * r205924;
        double r205975 = r205929 * r205929;
        double r205976 = r205974 - r205975;
        double r205977 = r205923 + r205928;
        double r205978 = fma(r205923, r205923, r205929);
        double r205979 = r205977 * r205978;
        double r205980 = r205976 / r205979;
        double r205981 = r205923 * r205980;
        double r205982 = r205981 / r205939;
        double r205983 = r205942 ? r205973 : r205982;
        double r205984 = r205922 ? r205940 : r205983;
        return r205984;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.000361777145262516

   1. Initial program 1.1

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

   2. Simplified 1.1

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

   4. Applied flip-- 1.1

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   8. Applied flip-- 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

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

   12. Applied add-exp-log 0.1

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

   if -0.000361777145262516 < x < 0.0005078281910445067

   1. Initial program 30.5

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

   2. Simplified 30.5

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

   3. Taylor expanded around 0 30.5

      \[\leadsto \color{blue}{\left(0.0078125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.125 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{2}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right) + 1\right)\right) - \left(0.09375 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{4}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right) + \sqrt{1} \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}\right)}\]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{3}}} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.0078125, \mathsf{fma}\left(0.125, \frac{\left(x \cdot x\right) \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}}{1}, 1 - \mathsf{fma}\left(\frac{0.09375}{1} \cdot \frac{{x}^{4}}{1}, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}} \cdot \sqrt{1}\right)\right)\right)}\]

   if 0.0005078281910445067 < x

   1. Initial program 1.1

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

   2. Simplified 1.1

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

   4. Applied flip-- 1.1

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

   5. Simplified 0.2

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

   6. Simplified 0.2

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

   8. Applied flip-- 0.2

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

   9. Simplified 0.2

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

   10. Simplified 0.2

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

   12. Applied flip-- 0.2

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

   13. Applied associate-/l/ 0.2

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

   14. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.617771452625160211354071737588355972548 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{\log \left(1 \cdot 1 - \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{\left(1 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 0.5} \cdot 1}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{elif}\;x \le 5.078281910445067021120379990861692931503 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{3}}} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.0078125, \mathsf{fma}\left(0.125, \frac{\sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \left(x \cdot x\right)}{1}, 1 - \mathsf{fma}\left(\frac{0.09375}{1} \cdot \frac{{x}^{4}}{1}, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \sqrt{1} \cdot \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{\left(1 + \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \mathsf{fma}\left(1, 1, \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))))))