3.9.2 imaginary part (p56)

Average Error: 0 → 0

Time: 29.8s

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 r1833525 = 1.0;
        double r1833526 = -r1833525;
        double r1833527 = 2.0;
        double r1833528 = r1833526 / r1833527;
        double r1833529 = 3.0;
        double r1833530 = sqrt(r1833529);
        double r1833531 = r1833530 / r1833527;
        double r1833532 = /* ERROR: no complex support in C */;
        double r1833533 = r1833532 * r1833532;
        double r1833534 = r1833533 * r1833532;
        double r1833535 = r1833534 * r1833532;
        double r1833536 = -r1833527;
        double r1833537 = 0.0;
        double r1833538 = /* ERROR: no complex support in C */;
        double r1833539 = r1833538 * r1833532;
        double r1833540 = r1833539 * r1833532;
        double r1833541 = r1833540 * r1833532;
        double r1833542 = r1833535 + r1833541;
        double r1833543 = 5.0;
        double r1833544 = /* ERROR: no complex support in C */;
        double r1833545 = r1833544 * r1833532;
        double r1833546 = r1833545 * r1833532;
        double r1833547 = r1833542 + r1833546;
        double r1833548 = 4.0;
        double r1833549 = /* ERROR: no complex support in C */;
        double r1833550 = r1833549 * r1833532;
        double r1833551 = r1833547 + r1833550;
        double r1833552 = 7.0;
        double r1833553 = /* ERROR: no complex support in C */;
        double r1833554 = r1833551 + r1833553;
        double r1833555 = /* ERROR: no complex support in C */;
        return r1833555;
}

↓

double f() {
        double r1833556 = 5.0;
        double r1833557 = 3.0;
        double r1833558 = sqrt(r1833557);
        double r1833559 = 2.0;
        double r1833560 = r1833558 / r1833559;
        double r1833561 = 4.0;
        double r1833562 = r1833560 * r1833561;
        double r1833563 = /* ERROR: no complex support in C */;
        double r1833564 = 0.25;
        double r1833565 = r1833560 * r1833560;
        double r1833566 = r1833564 - r1833565;
        double r1833567 = -r1833558;
        double r1833568 = r1833567 / r1833559;
        double r1833569 = /* ERROR: no complex support in C */;
        double r1833570 = 6.0;
        double r1833571 = -2.0;
        double r1833572 = r1833571 * r1833558;
        double r1833573 = r1833572 / r1833559;
        double r1833574 = /* ERROR: no complex support in C */;
        double r1833575 = r1833574 + r1833569;
        double r1833576 = r1833569 * r1833575;
        double r1833577 = r1833563 + r1833576;
        double r1833578 = /* ERROR: no complex support in C */;
        return r1833578;
}

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 2019104 +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))))