3.9.2 imaginary part (p56)

Average Error: 0 → 0

Time: 30.4s

Precision: 64

Math

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

↓

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

C

double f() {
        double r2498727 = 1.0;
        double r2498728 = -r2498727;
        double r2498729 = 2.0;
        double r2498730 = r2498728 / r2498729;
        double r2498731 = 3.0;
        double r2498732 = sqrt(r2498731);
        double r2498733 = r2498732 / r2498729;
        double r2498734 = /* ERROR: no complex support in C */;
        double r2498735 = r2498734 * r2498734;
        double r2498736 = r2498735 * r2498734;
        double r2498737 = r2498736 * r2498734;
        double r2498738 = -r2498729;
        double r2498739 = 0.0;
        double r2498740 = /* ERROR: no complex support in C */;
        double r2498741 = r2498740 * r2498734;
        double r2498742 = r2498741 * r2498734;
        double r2498743 = r2498742 * r2498734;
        double r2498744 = r2498737 + r2498743;
        double r2498745 = 5.0;
        double r2498746 = /* ERROR: no complex support in C */;
        double r2498747 = r2498746 * r2498734;
        double r2498748 = r2498747 * r2498734;
        double r2498749 = r2498744 + r2498748;
        double r2498750 = 4.0;
        double r2498751 = /* ERROR: no complex support in C */;
        double r2498752 = r2498751 * r2498734;
        double r2498753 = r2498749 + r2498752;
        double r2498754 = 7.0;
        double r2498755 = /* ERROR: no complex support in C */;
        double r2498756 = r2498753 + r2498755;
        double r2498757 = /* ERROR: no complex support in C */;
        return r2498757;
}

↓

double f() {
        double r2498758 = 5.0;
        double r2498759 = 3.0;
        double r2498760 = sqrt(r2498759);
        double r2498761 = 2.0;
        double r2498762 = r2498760 / r2498761;
        double r2498763 = 4.0;
        double r2498764 = r2498762 * r2498763;
        double r2498765 = /* ERROR: no complex support in C */;
        double r2498766 = 0.25;
        double r2498767 = r2498762 * r2498762;
        double r2498768 = r2498766 - r2498767;
        double r2498769 = -r2498760;
        double r2498770 = r2498769 / r2498761;
        double r2498771 = /* ERROR: no complex support in C */;
        double r2498772 = 6.0;
        double r2498773 = -2.0;
        double r2498774 = r2498773 * r2498760;
        double r2498775 = r2498774 / r2498761;
        double r2498776 = /* ERROR: no complex support in C */;
        double r2498777 = r2498776 + r2498771;
        double r2498778 = r2498771 * r2498777;
        double r2498779 = r2498765 + r2498778;
        double r2498780 = /* ERROR: no complex support in C */;
        return r2498780;
}

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

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

3. Final simplification 0

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

Reproduce

herbie shell --seed 2019107 +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))))