3.9.1 imaginary part (p56)

Average Error: 0 → 0

Time: 24.3s

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 r1914709 = -1.0;
        double r1914710 = 1.0;
        double r1914711 = /* ERROR: no complex support in C */;
        double r1914712 = r1914711 * r1914711;
        double r1914713 = r1914712 * r1914711;
        double r1914714 = r1914713 * r1914711;
        double r1914715 = r1914714 * r1914711;
        double r1914716 = r1914715 * r1914711;
        double r1914717 = 6.0;
        double r1914718 = 0.0;
        double r1914719 = /* ERROR: no complex support in C */;
        double r1914720 = r1914719 * r1914711;
        double r1914721 = r1914720 * r1914711;
        double r1914722 = r1914721 * r1914711;
        double r1914723 = r1914722 * r1914711;
        double r1914724 = r1914723 * r1914711;
        double r1914725 = r1914716 + r1914724;
        double r1914726 = 15.0;
        double r1914727 = /* ERROR: no complex support in C */;
        double r1914728 = r1914727 * r1914711;
        double r1914729 = r1914728 * r1914711;
        double r1914730 = r1914729 * r1914711;
        double r1914731 = r1914730 * r1914711;
        double r1914732 = r1914725 + r1914731;
        double r1914733 = 20.0;
        double r1914734 = /* ERROR: no complex support in C */;
        double r1914735 = r1914734 * r1914711;
        double r1914736 = r1914735 * r1914711;
        double r1914737 = r1914736 * r1914711;
        double r1914738 = r1914732 + r1914737;
        double r1914739 = r1914738 + r1914729;
        double r1914740 = r1914739 + r1914720;
        double r1914741 = /* ERROR: no complex support in C */;
        double r1914742 = r1914740 + r1914741;
        double r1914743 = /* ERROR: no complex support in C */;
        return r1914743;
}

↓

double f() {
        double r1914744 = -1.0;
        double r1914745 = 1.0;
        double r1914746 = r1914744 + r1914745;
        double r1914747 = r1914744 - r1914745;
        double r1914748 = r1914746 * r1914747;
        double r1914749 = 20.0;
        double r1914750 = r1914749 * r1914744;
        double r1914751 = 0.0;
        double r1914752 = r1914751 * r1914745;
        double r1914753 = r1914750 - r1914752;
        double r1914754 = r1914748 * r1914753;
        double r1914755 = r1914744 + r1914744;
        double r1914756 = r1914755 * r1914745;
        double r1914757 = r1914744 * r1914751;
        double r1914758 = fma(r1914749, r1914745, r1914757);
        double r1914759 = r1914756 * r1914758;
        double r1914760 = r1914754 - r1914759;
        double r1914761 = r1914758 * r1914748;
        double r1914762 = fma(r1914753, r1914756, r1914761);
        double r1914763 = /* ERROR: no complex support in C */;
        double r1914764 = 6.0;
        double r1914765 = 15.0;
        double r1914766 = r1914744 * r1914765;
        double r1914767 = r1914766 - r1914752;
        double r1914768 = r1914764 + r1914767;
        double r1914769 = fma(r1914745, r1914765, r1914757);
        double r1914770 = r1914769 + r1914751;
        double r1914771 = /* ERROR: no complex support in C */;
        double r1914772 = /* ERROR: no complex support in C */;
        double r1914773 = /* ERROR: no complex support in C */;
        double r1914774 = /* ERROR: no complex support in C */;
        double r1914775 = /* ERROR: no complex support in C */;
        double r1914776 = r1914774 + r1914775;
        double r1914777 = r1914772 * r1914776;
        double r1914778 = r1914773 + r1914777;
        double r1914779 = r1914772 * r1914778;
        double r1914780 = r1914771 + r1914779;
        double r1914781 = r1914780 * r1914775;
        double r1914782 = /* ERROR: no complex support in C */;
        double r1914783 = r1914781 + r1914782;
        double r1914784 = r1914763 + r1914783;
        double r1914785 = /* ERROR: no complex support in C */;
        return r1914785;
}

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 2019192 +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))))