3.9.2 imaginary part (p56)

Average Error: 0 → 0

Time: 18.3s

Precision: 64

Math

\[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]

↓

\[\Im(\left(\left(\frac{-1}{2} + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right) \cdot \left(\left(\frac{-1}{2} + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right) + \left(6 + \left(-\sqrt{3}\right) i\right)\right) + \left(5 + \sqrt{3} \cdot 2 i\right)\right))\]

C

double f() {
        double r905805 = 1.0;
        double r905806 = -r905805;
        double r905807 = 2.0;
        double r905808 = r905806 / r905807;
        double r905809 = 3.0;
        double r905810 = sqrt(r905809);
        double r905811 = r905810 / r905807;
        double r905812 = /* ERROR: no complex support in C */;
        double r905813 = r905812 * r905812;
        double r905814 = r905813 * r905812;
        double r905815 = r905814 * r905812;
        double r905816 = -r905807;
        double r905817 = 0.0;
        double r905818 = /* ERROR: no complex support in C */;
        double r905819 = r905818 * r905812;
        double r905820 = r905819 * r905812;
        double r905821 = r905820 * r905812;
        double r905822 = r905815 + r905821;
        double r905823 = 5.0;
        double r905824 = /* ERROR: no complex support in C */;
        double r905825 = r905824 * r905812;
        double r905826 = r905825 * r905812;
        double r905827 = r905822 + r905826;
        double r905828 = 4.0;
        double r905829 = /* ERROR: no complex support in C */;
        double r905830 = r905829 * r905812;
        double r905831 = r905827 + r905830;
        double r905832 = 7.0;
        double r905833 = /* ERROR: no complex support in C */;
        double r905834 = r905831 + r905833;
        double r905835 = /* ERROR: no complex support in C */;
        return r905835;
}

↓

double f() {
        double r905836 = -0.5;
        double r905837 = 3.0;
        double r905838 = sqrt(r905837);
        double r905839 = -4.0;
        double r905840 = r905838 / r905839;
        double r905841 = r905840 + r905840;
        double r905842 = /* ERROR: no complex support in C */;
        double r905843 = 6.0;
        double r905844 = -r905838;
        double r905845 = /* ERROR: no complex support in C */;
        double r905846 = r905842 + r905845;
        double r905847 = r905842 * r905846;
        double r905848 = 5.0;
        double r905849 = 2.0;
        double r905850 = r905838 * r905849;
        double r905851 = /* ERROR: no complex support in C */;
        double r905852 = r905847 + r905851;
        double r905853 = /* ERROR: no complex support in C */;
        return r905853;
}

Derivation

1. Initial program 0

   \[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) + \left(\left(\left(\left(\left(-2\right) + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(\left(5 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(\left(4 + 0 i\right) \cdot \left(\frac{-1}{2} + \frac{\sqrt{3}}{2} i\right)\right)\right) + \left(7 + 0 i\right)\right))\]

2. Simplified 0

   \[\leadsto \color{blue}{\Im(\left(\left(5 + 2 \cdot \sqrt{3} i\right) + \left(\left(\left(\frac{1}{4} - \frac{\frac{3}{2}}{2}\right) + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right) \cdot \left(\left(6 + -1 \cdot \sqrt{3} i\right) + \left(\left(\frac{1}{4} - \frac{\frac{3}{2}}{2}\right) + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right)\right)\right)\right))}\]

3. Final simplification 0

   \[\leadsto \Im(\left(\left(\frac{-1}{2} + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right) \cdot \left(\left(\frac{-1}{2} + \left(\frac{\sqrt{3}}{-4} + \frac{\sqrt{3}}{-4}\right) i\right) + \left(6 + \left(-\sqrt{3}\right) i\right)\right) + \left(5 + \sqrt{3} \cdot 2 i\right)\right))\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore ()
  :name "3.9.2 imaginary part (p56)"
  (im (+.c (+.c (+.c (+.c (*.c (*.c (*.c (complex (/ (- 1) 2) (/ (sqrt 3) 2)) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (*.c (*.c (*.c (complex (- 2) 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (*.c (complex 5 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2))) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (*.c (complex 4 0) (complex (/ (- 1) 2) (/ (sqrt 3) 2)))) (complex 7 0))))