3.9.2 imaginary part (p56)

Average Error: 0 → 0

Time: 21.5s

Precision: 64

Math

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

↓

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

C

double f() {
        double r830637 = 1.0;
        double r830638 = -r830637;
        double r830639 = 2.0;
        double r830640 = r830638 / r830639;
        double r830641 = 3.0;
        double r830642 = sqrt(r830641);
        double r830643 = r830642 / r830639;
        double r830644 = /* ERROR: no complex support in C */;
        double r830645 = r830644 * r830644;
        double r830646 = r830645 * r830644;
        double r830647 = r830646 * r830644;
        double r830648 = -r830639;
        double r830649 = 0.0;
        double r830650 = /* ERROR: no complex support in C */;
        double r830651 = r830650 * r830644;
        double r830652 = r830651 * r830644;
        double r830653 = r830652 * r830644;
        double r830654 = r830647 + r830653;
        double r830655 = 5.0;
        double r830656 = /* ERROR: no complex support in C */;
        double r830657 = r830656 * r830644;
        double r830658 = r830657 * r830644;
        double r830659 = r830654 + r830658;
        double r830660 = 4.0;
        double r830661 = /* ERROR: no complex support in C */;
        double r830662 = r830661 * r830644;
        double r830663 = r830659 + r830662;
        double r830664 = 7.0;
        double r830665 = /* ERROR: no complex support in C */;
        double r830666 = r830663 + r830665;
        double r830667 = /* ERROR: no complex support in C */;
        return r830667;
}

↓

double f() {
        double r830668 = 5.0;
        double r830669 = 3.0;
        double r830670 = sqrt(r830669);
        double r830671 = 4.0;
        double r830672 = r830670 * r830671;
        double r830673 = 2.0;
        double r830674 = r830672 / r830673;
        double r830675 = /* ERROR: no complex support in C */;
        double r830676 = 0.25;
        double r830677 = r830670 / r830673;
        double r830678 = r830677 * r830677;
        double r830679 = -r830678;
        double r830680 = r830676 + r830679;
        double r830681 = -r830677;
        double r830682 = /* ERROR: no complex support in C */;
        double r830683 = 6.0;
        double r830684 = -2.0;
        double r830685 = r830677 * r830684;
        double r830686 = /* ERROR: no complex support in C */;
        double r830687 = r830682 + r830686;
        double r830688 = r830682 * r830687;
        double r830689 = r830675 + r830688;
        double r830690 = /* ERROR: no complex support in C */;
        return r830690;
}

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

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

3. Final simplification 0

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

Reproduce

herbie shell --seed 2019163 +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))))