3.9.1 imaginary part (p56)

Average Error: 0 → 0

Time: 23.5s

Precision: 64

Math

\[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(-1 + 1 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(\left(\left(\left(\left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(20 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(1 + 0.0 i\right)\right))\]

↓

\[\Im(\left(\left(\left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right)\right) \cdot \left(20 \cdot -1 - 0.0 \cdot 1\right) - \left(\left(-1 + -1\right) \cdot 1\right) \cdot \mathsf{fma}\left(20, 1, -1 \cdot 0.0\right)\right) + \mathsf{fma}\left(20 \cdot -1 - 0.0 \cdot 1, \left(-1 + -1\right) \cdot 1, \mathsf{fma}\left(20, 1, -1 \cdot 0.0\right) \cdot \left(\left(-1 + 1\right) \cdot \left(-1 - 1\right)\right)\right) i\right) + \left(\left(\left(\left(\left(6 + \left(-1 \cdot 15 - 0.0 \cdot 1\right)\right) + \left(\mathsf{fma}\left(1, 15, -1 \cdot 0.0\right) + 0.0\right) i\right) + \left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right) + \left(-1 + -1\right) \cdot 1 i\right) \cdot \left(\left(\left(-1 \cdot 15 - 0.0 \cdot 1\right) + \mathsf{fma}\left(1, 15, -1 \cdot 0.0\right) i\right) + \left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right) + \left(-1 + -1\right) \cdot 1 i\right) \cdot \left(\left(6 + 0.0 i\right) + \left(-1 + 1 i\right)\right)\right)\right)\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(1 + 0.0 i\right)\right)\right))\]

C

double f() {
        double r1981343 = -1.0;
        double r1981344 = 1.0;
        double r1981345 = /* ERROR: no complex support in C */;
        double r1981346 = r1981345 * r1981345;
        double r1981347 = r1981346 * r1981345;
        double r1981348 = r1981347 * r1981345;
        double r1981349 = r1981348 * r1981345;
        double r1981350 = r1981349 * r1981345;
        double r1981351 = 6.0;
        double r1981352 = 0.0;
        double r1981353 = /* ERROR: no complex support in C */;
        double r1981354 = r1981353 * r1981345;
        double r1981355 = r1981354 * r1981345;
        double r1981356 = r1981355 * r1981345;
        double r1981357 = r1981356 * r1981345;
        double r1981358 = r1981357 * r1981345;
        double r1981359 = r1981350 + r1981358;
        double r1981360 = 15.0;
        double r1981361 = /* ERROR: no complex support in C */;
        double r1981362 = r1981361 * r1981345;
        double r1981363 = r1981362 * r1981345;
        double r1981364 = r1981363 * r1981345;
        double r1981365 = r1981364 * r1981345;
        double r1981366 = r1981359 + r1981365;
        double r1981367 = 20.0;
        double r1981368 = /* ERROR: no complex support in C */;
        double r1981369 = r1981368 * r1981345;
        double r1981370 = r1981369 * r1981345;
        double r1981371 = r1981370 * r1981345;
        double r1981372 = r1981366 + r1981371;
        double r1981373 = r1981372 + r1981363;
        double r1981374 = r1981373 + r1981354;
        double r1981375 = /* ERROR: no complex support in C */;
        double r1981376 = r1981374 + r1981375;
        double r1981377 = /* ERROR: no complex support in C */;
        return r1981377;
}

↓

double f() {
        double r1981378 = -1.0;
        double r1981379 = 1.0;
        double r1981380 = r1981378 + r1981379;
        double r1981381 = r1981378 - r1981379;
        double r1981382 = r1981380 * r1981381;
        double r1981383 = 20.0;
        double r1981384 = r1981383 * r1981378;
        double r1981385 = 0.0;
        double r1981386 = r1981385 * r1981379;
        double r1981387 = r1981384 - r1981386;
        double r1981388 = r1981382 * r1981387;
        double r1981389 = r1981378 + r1981378;
        double r1981390 = r1981389 * r1981379;
        double r1981391 = r1981378 * r1981385;
        double r1981392 = fma(r1981383, r1981379, r1981391);
        double r1981393 = r1981390 * r1981392;
        double r1981394 = r1981388 - r1981393;
        double r1981395 = r1981392 * r1981382;
        double r1981396 = fma(r1981387, r1981390, r1981395);
        double r1981397 = /* ERROR: no complex support in C */;
        double r1981398 = 6.0;
        double r1981399 = 15.0;
        double r1981400 = r1981378 * r1981399;
        double r1981401 = r1981400 - r1981386;
        double r1981402 = r1981398 + r1981401;
        double r1981403 = fma(r1981379, r1981399, r1981391);
        double r1981404 = r1981403 + r1981385;
        double r1981405 = /* ERROR: no complex support in C */;
        double r1981406 = /* ERROR: no complex support in C */;
        double r1981407 = /* ERROR: no complex support in C */;
        double r1981408 = /* ERROR: no complex support in C */;
        double r1981409 = /* ERROR: no complex support in C */;
        double r1981410 = r1981408 + r1981409;
        double r1981411 = r1981406 * r1981410;
        double r1981412 = r1981407 + r1981411;
        double r1981413 = r1981406 * r1981412;
        double r1981414 = r1981405 + r1981413;
        double r1981415 = r1981414 * r1981409;
        double r1981416 = /* ERROR: no complex support in C */;
        double r1981417 = r1981415 + r1981416;
        double r1981418 = r1981397 + r1981417;
        double r1981419 = /* ERROR: no complex support in C */;
        return r1981419;
}

Derivation

1. Initial program 0

   \[\Im(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(-1 + 1 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(\left(\left(\left(\left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(\left(20 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(\left(15 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(\left(6 + 0.0 i\right) \cdot \left(-1 + 1 i\right)\right)\right) + \left(1 + 0.0 i\right)\right))\]

2. Simplified 0

   \[\leadsto \color{blue}{\Im(\left(\left(\left(\left(\left(1 + -1\right) \cdot \left(-1 - 1\right)\right) \cdot \left(-1 \cdot 20 - 1 \cdot 0.0\right) - \left(1 \cdot \left(-1 + -1\right)\right) \cdot \mathsf{fma}\left(20, 1, 0.0 \cdot -1\right)\right) + \mathsf{fma}\left(-1 \cdot 20 - 1 \cdot 0.0, 1 \cdot \left(-1 + -1\right), \mathsf{fma}\left(20, 1, 0.0 \cdot -1\right) \cdot \left(\left(1 + -1\right) \cdot \left(-1 - 1\right)\right)\right) i\right) + \left(\left(\left(-1 + 1 i\right) \cdot \left(\left(\left(\left(1 + -1\right) \cdot \left(-1 - 1\right) + 1 \cdot \left(-1 + -1\right) i\right) \cdot \left(\left(\left(-1 \cdot 15 - 1 \cdot 0.0\right) + \mathsf{fma}\left(1, 15, 0.0 \cdot -1\right) i\right) + \left(\left(\left(1 + -1\right) \cdot \left(-1 - 1\right) + 1 \cdot \left(-1 + -1\right) i\right) \cdot \left(\left(-1 + 1 i\right) + \left(6 + 0.0 i\right)\right)\right)\right)\right) + \left(\left(\left(-1 \cdot 15 - 1 \cdot 0.0\right) + 6\right) + \left(\mathsf{fma}\left(1, 15, 0.0 \cdot -1\right) + 0.0\right) i\right)\right)\right) + \left(1 + 0.0 i\right)\right)\right))}\]

3. Final simplification 0

   \[\leadsto \Im(\left(\left(\left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right)\right) \cdot \left(20 \cdot -1 - 0.0 \cdot 1\right) - \left(\left(-1 + -1\right) \cdot 1\right) \cdot \mathsf{fma}\left(20, 1, -1 \cdot 0.0\right)\right) + \mathsf{fma}\left(20 \cdot -1 - 0.0 \cdot 1, \left(-1 + -1\right) \cdot 1, \mathsf{fma}\left(20, 1, -1 \cdot 0.0\right) \cdot \left(\left(-1 + 1\right) \cdot \left(-1 - 1\right)\right)\right) i\right) + \left(\left(\left(\left(\left(6 + \left(-1 \cdot 15 - 0.0 \cdot 1\right)\right) + \left(\mathsf{fma}\left(1, 15, -1 \cdot 0.0\right) + 0.0\right) i\right) + \left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right) + \left(-1 + -1\right) \cdot 1 i\right) \cdot \left(\left(\left(-1 \cdot 15 - 0.0 \cdot 1\right) + \mathsf{fma}\left(1, 15, -1 \cdot 0.0\right) i\right) + \left(\left(\left(-1 + 1\right) \cdot \left(-1 - 1\right) + \left(-1 + -1\right) \cdot 1 i\right) \cdot \left(\left(6 + 0.0 i\right) + \left(-1 + 1 i\right)\right)\right)\right)\right)\right) \cdot \left(-1 + 1 i\right)\right) + \left(1 + 0.0 i\right)\right)\right))\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore ()
  :name "3.9.1 imaginary part (p56)"
  (im (+ (+ (+ (+ (+ (+ (* (* (* (* (* (complex -1.0 1.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (* (* (* (* (* (complex 6.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (* (* (complex 15.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (* (complex 20.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (* (complex 15.0 0.0) (complex -1.0 1.0)) (complex -1.0 1.0))) (* (complex 6.0 0.0) (complex -1.0 1.0))) (complex 1.0 0.0))))