3.9.1 imaginary part (p56)

Average Error: 0 → 0

Time: 24.8s

Precision: 64

Math

\[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(-1 + 1 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(\left(\left(\left(\left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(20 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(1 + 0.0 i\right)\right))\]

↓

\[\Im(\left(\left(\left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(1 + 0.0 i\right)\right) + \left(\left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \left(\left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) \cdot \left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right)\right)\right)\right) + \left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right) \cdot \left(\left(\left(15 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - 0.0 \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \mathsf{fma}\left(15, 1 \cdot -1 + 1 \cdot -1, 0.0 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) + \left(\left(\left(20 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(20, 1, 0.0 \cdot -1\right) i\right) + \left(15 + 0.0 i\right)\right)\right)\right)\right))\]

C

double f() {
        double r3366852 = -1.0;
        double r3366853 = 1.0;
        double r3366854 = /* ERROR: no complex support in C */;
        double r3366855 = r3366854 * r3366854;
        double r3366856 = r3366855 * r3366854;
        double r3366857 = r3366856 * r3366854;
        double r3366858 = r3366857 * r3366854;
        double r3366859 = r3366858 * r3366854;
        double r3366860 = 6.0;
        double r3366861 = 0.0;
        double r3366862 = /* ERROR: no complex support in C */;
        double r3366863 = r3366862 * r3366854;
        double r3366864 = r3366863 * r3366854;
        double r3366865 = r3366864 * r3366854;
        double r3366866 = r3366865 * r3366854;
        double r3366867 = r3366866 * r3366854;
        double r3366868 = r3366859 + r3366867;
        double r3366869 = 15.0;
        double r3366870 = /* ERROR: no complex support in C */;
        double r3366871 = r3366870 * r3366854;
        double r3366872 = r3366871 * r3366854;
        double r3366873 = r3366872 * r3366854;
        double r3366874 = r3366873 * r3366854;
        double r3366875 = r3366868 + r3366874;
        double r3366876 = 20.0;
        double r3366877 = /* ERROR: no complex support in C */;
        double r3366878 = r3366877 * r3366854;
        double r3366879 = r3366878 * r3366854;
        double r3366880 = r3366879 * r3366854;
        double r3366881 = r3366875 + r3366880;
        double r3366882 = r3366881 + r3366872;
        double r3366883 = r3366882 + r3366863;
        double r3366884 = /* ERROR: no complex support in C */;
        double r3366885 = r3366883 + r3366884;
        double r3366886 = /* ERROR: no complex support in C */;
        return r3366886;
}

↓

double f() {
        double r3366887 = 6.0;
        double r3366888 = -1.0;
        double r3366889 = r3366887 * r3366888;
        double r3366890 = 0.0;
        double r3366891 = 1.0;
        double r3366892 = r3366890 * r3366891;
        double r3366893 = r3366889 - r3366892;
        double r3366894 = r3366890 * r3366888;
        double r3366895 = fma(r3366887, r3366891, r3366894);
        double r3366896 = /* ERROR: no complex support in C */;
        double r3366897 = /* ERROR: no complex support in C */;
        double r3366898 = r3366896 + r3366897;
        double r3366899 = r3366888 * r3366888;
        double r3366900 = r3366891 * r3366891;
        double r3366901 = r3366899 - r3366900;
        double r3366902 = r3366901 * r3366901;
        double r3366903 = r3366891 * r3366888;
        double r3366904 = r3366903 + r3366903;
        double r3366905 = r3366904 * r3366904;
        double r3366906 = r3366902 - r3366905;
        double r3366907 = r3366904 * r3366901;
        double r3366908 = r3366907 + r3366907;
        double r3366909 = /* ERROR: no complex support in C */;
        double r3366910 = /* ERROR: no complex support in C */;
        double r3366911 = r3366896 + r3366910;
        double r3366912 = r3366909 * r3366911;
        double r3366913 = r3366898 + r3366912;
        double r3366914 = 15.0;
        double r3366915 = r3366914 * r3366901;
        double r3366916 = r3366890 * r3366904;
        double r3366917 = r3366915 - r3366916;
        double r3366918 = r3366890 * r3366901;
        double r3366919 = fma(r3366914, r3366904, r3366918);
        double r3366920 = /* ERROR: no complex support in C */;
        double r3366921 = 20.0;
        double r3366922 = r3366921 * r3366888;
        double r3366923 = r3366922 - r3366892;
        double r3366924 = fma(r3366921, r3366891, r3366894);
        double r3366925 = /* ERROR: no complex support in C */;
        double r3366926 = /* ERROR: no complex support in C */;
        double r3366927 = r3366925 + r3366926;
        double r3366928 = r3366920 + r3366927;
        double r3366929 = r3366910 * r3366928;
        double r3366930 = r3366913 + r3366929;
        double r3366931 = /* ERROR: no complex support in C */;
        return r3366931;
}

Derivation

1. Initial program 0

   \[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(-1 + 1 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(\left(\left(\left(\left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(20 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(1 + 0.0 i\right)\right))\]

2. Simplified 0

   \[\leadsto \color{blue}{\Im(\left(\left(\left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(1 + 0.0 i\right)\right) + \left(\left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \left(\left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) \cdot \left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right)\right)\right)\right) + \left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right) \cdot \left(\left(\left(15 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - 0.0 \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \mathsf{fma}\left(15, 1 \cdot -1 + 1 \cdot -1, 0.0 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) + \left(\left(\left(20 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(20, 1, 0.0 \cdot -1\right) i\right) + \left(15 + 0.0 i\right)\right)\right)\right)\right))}\]

3. Final simplification 0

   \[\leadsto \Im(\left(\left(\left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(1 + 0.0 i\right)\right) + \left(\left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \left(\left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) \cdot \left(\left(\left(6 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(6, 1, 0.0 \cdot -1\right) i\right) + \left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right)\right)\right)\right) + \left(\left(\left(-1 \cdot -1 - 1 \cdot 1\right) + \left(1 \cdot -1 + 1 \cdot -1\right) i\right) \cdot \left(\left(\left(15 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right) - 0.0 \cdot \left(1 \cdot -1 + 1 \cdot -1\right)\right) + \mathsf{fma}\left(15, 1 \cdot -1 + 1 \cdot -1, 0.0 \cdot \left(-1 \cdot -1 - 1 \cdot 1\right)\right) i\right) + \left(\left(\left(20 \cdot -1 - 0.0 \cdot 1\right) + \mathsf{fma}\left(20, 1, 0.0 \cdot -1\right) i\right) + \left(15 + 0.0 i\right)\right)\right)\right)\right))\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore ()
  :name "3.9.1 imaginary part (p56)"
  (im (+ (+ (+ (+ (+ (+ (* (* (* (* (* (complex -1.0 1.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (* (* (* (* (* (complex 6.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (* (* (complex 15.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (* (complex 20.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (complex 15.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (complex 6.0 0.0) (complex -1.0 1.0))) (complex 1.0 0.0))))