3.9.2 imaginary part (p56)

Average Error: 0 → 0

Time: 41.1s

Precision: 64

Math

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

↓

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

C

double f() {
        double r4183303 = 1.0;
        double r4183304 = -r4183303;
        double r4183305 = 2.0;
        double r4183306 = r4183304 / r4183305;
        double r4183307 = 3.0;
        double r4183308 = sqrt(r4183307);
        double r4183309 = r4183308 / r4183305;
        double r4183310 = /* ERROR: no complex support in C */;
        double r4183311 = r4183310 * r4183310;
        double r4183312 = r4183311 * r4183310;
        double r4183313 = r4183312 * r4183310;
        double r4183314 = -r4183305;
        double r4183315 = 0.0;
        double r4183316 = /* ERROR: no complex support in C */;
        double r4183317 = r4183316 * r4183310;
        double r4183318 = r4183317 * r4183310;
        double r4183319 = r4183318 * r4183310;
        double r4183320 = r4183313 + r4183319;
        double r4183321 = 5.0;
        double r4183322 = /* ERROR: no complex support in C */;
        double r4183323 = r4183322 * r4183310;
        double r4183324 = r4183323 * r4183310;
        double r4183325 = r4183320 + r4183324;
        double r4183326 = 4.0;
        double r4183327 = /* ERROR: no complex support in C */;
        double r4183328 = r4183327 * r4183310;
        double r4183329 = r4183325 + r4183328;
        double r4183330 = 7.0;
        double r4183331 = /* ERROR: no complex support in C */;
        double r4183332 = r4183329 + r4183331;
        double r4183333 = /* ERROR: no complex support in C */;
        return r4183333;
}

↓

double f() {
        double r4183334 = 5.0;
        double r4183335 = 3.0;
        double r4183336 = sqrt(r4183335);
        double r4183337 = 2.0;
        double r4183338 = r4183336 / r4183337;
        double r4183339 = 4.0;
        double r4183340 = r4183338 * r4183339;
        double r4183341 = /* ERROR: no complex support in C */;
        double r4183342 = 0.25;
        double r4183343 = r4183338 * r4183338;
        double r4183344 = r4183342 - r4183343;
        double r4183345 = -4.0;
        double r4183346 = r4183336 / r4183345;
        double r4183347 = r4183346 + r4183346;
        double r4183348 = /* ERROR: no complex support in C */;
        double r4183349 = 6.0;
        double r4183350 = -2.0;
        double r4183351 = r4183350 * r4183338;
        double r4183352 = /* ERROR: no complex support in C */;
        double r4183353 = r4183348 + r4183352;
        double r4183354 = r4183348 * r4183353;
        double r4183355 = r4183341 + r4183354;
        double r4183356 = /* ERROR: no complex support in C */;
        return r4183356;
}

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

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

3. Final simplification 0

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

Reproduce

herbie shell --seed 2019128 +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))))