math.cos on complex, imaginary part

Average Error: 43.2 → 0.9

Time: 33.3s

Precision: 64

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

↓

\[\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\frac{\sqrt{\mathsf{fma}\left(\left(\frac{1}{27} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right), \left(im \cdot im\right), 8\right)} \cdot \left(im \cdot \sqrt{4 - \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)}\right)}{\sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \frac{-1}{3}, 2\right)} \cdot \sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \left(\mathsf{fma}\left(\left(im \cdot im\right), \frac{1}{9}, \frac{-2}{3}\right)\right), 4\right)}}\right)\right) \cdot \left(\sin re \cdot \left(-0.5\right)\right)\]

C

double f(double re, double im) {
        double r5943744 = 0.5;
        double r5943745 = re;
        double r5943746 = sin(r5943745);
        double r5943747 = r5943744 * r5943746;
        double r5943748 = im;
        double r5943749 = -r5943748;
        double r5943750 = exp(r5943749);
        double r5943751 = exp(r5943748);
        double r5943752 = r5943750 - r5943751;
        double r5943753 = r5943747 * r5943752;
        return r5943753;
}

↓

double f(double re, double im) {
        double r5943754 = 0.016666666666666666;
        double r5943755 = im;
        double r5943756 = 5.0;
        double r5943757 = pow(r5943755, r5943756);
        double r5943758 = 0.037037037037037035;
        double r5943759 = r5943755 * r5943755;
        double r5943760 = r5943759 * r5943759;
        double r5943761 = r5943758 * r5943760;
        double r5943762 = 8.0;
        double r5943763 = fma(r5943761, r5943759, r5943762);
        double r5943764 = sqrt(r5943763);
        double r5943765 = 4.0;
        double r5943766 = 0.3333333333333333;
        double r5943767 = r5943759 * r5943766;
        double r5943768 = r5943767 * r5943767;
        double r5943769 = r5943765 - r5943768;
        double r5943770 = sqrt(r5943769);
        double r5943771 = r5943755 * r5943770;
        double r5943772 = r5943764 * r5943771;
        double r5943773 = -0.3333333333333333;
        double r5943774 = 2.0;
        double r5943775 = fma(r5943759, r5943773, r5943774);
        double r5943776 = sqrt(r5943775);
        double r5943777 = 0.1111111111111111;
        double r5943778 = -0.6666666666666666;
        double r5943779 = fma(r5943759, r5943777, r5943778);
        double r5943780 = fma(r5943759, r5943779, r5943765);
        double r5943781 = sqrt(r5943780);
        double r5943782 = r5943776 * r5943781;
        double r5943783 = r5943772 / r5943782;
        double r5943784 = fma(r5943754, r5943757, r5943783);
        double r5943785 = re;
        double r5943786 = sin(r5943785);
        double r5943787 = 0.5;
        double r5943788 = -r5943787;
        double r5943789 = r5943786 * r5943788;
        double r5943790 = r5943784 * r5943789;
        return r5943790;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.2
Target	0.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.2

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

2. Taylor expanded around 0 0.8

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

3. Simplified 0.8

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(im \cdot \left(2 + \frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)\right)\right)}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 1.5

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(im \cdot \color{blue}{\left(\sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)} \cdot \sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)}\right)}\right)\right)\right)\]

6. Applied associate-*r* 1.3

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \color{blue}{\left(\left(im \cdot \sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)}\right) \cdot \sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)}\right)}\right)\right)\]
7. Using strategy rm

8. Applied flip3-+ 1.3

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\left(im \cdot \sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)}\right) \cdot \sqrt{\color{blue}{\frac{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}}\right)\right)\right)\]

9. Applied sqrt-div 1.3

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\left(im \cdot \sqrt{2 + \frac{1}{3} \cdot \left(im \cdot im\right)}\right) \cdot \color{blue}{\frac{\sqrt{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}}\right)\right)\right)\]

10. Applied flip-+ 1.3

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\left(im \cdot \sqrt{\color{blue}{\frac{2 \cdot 2 - \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}{2 - \frac{1}{3} \cdot \left(im \cdot im\right)}}}\right) \cdot \frac{\sqrt{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}\right)\right)\right)\]

11. Applied sqrt-div 0.9

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\left(im \cdot \color{blue}{\frac{\sqrt{2 \cdot 2 - \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}}{\sqrt{2 - \frac{1}{3} \cdot \left(im \cdot im\right)}}}\right) \cdot \frac{\sqrt{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}\right)\right)\right)\]

12. Applied associate-*r/ 0.9

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\color{blue}{\frac{im \cdot \sqrt{2 \cdot 2 - \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}}{\sqrt{2 - \frac{1}{3} \cdot \left(im \cdot im\right)}}} \cdot \frac{\sqrt{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}}{\sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}\right)\right)\right)\]

13. Applied frac-times 0.9

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \color{blue}{\left(\frac{\left(im \cdot \sqrt{2 \cdot 2 - \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}\right) \cdot \sqrt{{2}^{3} + {\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)}^{3}}}{\sqrt{2 - \frac{1}{3} \cdot \left(im \cdot im\right)} \cdot \sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}\right)}\right)\right)\]

14. Simplified 0.9

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\frac{\color{blue}{\sqrt{\mathsf{fma}\left(\left(\frac{1}{27} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right), \left(im \cdot im\right), 8\right)} \cdot \left(im \cdot \sqrt{4 - \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)}\right)}}{\sqrt{2 - \frac{1}{3} \cdot \left(im \cdot im\right)} \cdot \sqrt{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(im \cdot im\right)\right)\right)}}\right)\right)\right)\]

15. Simplified 0.9

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\frac{\sqrt{\mathsf{fma}\left(\left(\frac{1}{27} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right), \left(im \cdot im\right), 8\right)} \cdot \left(im \cdot \sqrt{4 - \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)}\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \left(\mathsf{fma}\left(\left(im \cdot im\right), \frac{1}{9}, \frac{-2}{3}\right)\right), 4\right)} \cdot \sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \frac{-1}{3}, 2\right)}}}\right)\right)\right)\]

16. Final simplification 0.9

    \[\leadsto \mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(\frac{\sqrt{\mathsf{fma}\left(\left(\frac{1}{27} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right), \left(im \cdot im\right), 8\right)} \cdot \left(im \cdot \sqrt{4 - \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)}\right)}{\sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \frac{-1}{3}, 2\right)} \cdot \sqrt{\mathsf{fma}\left(\left(im \cdot im\right), \left(\mathsf{fma}\left(\left(im \cdot im\right), \frac{1}{9}, \frac{-2}{3}\right)\right), 4\right)}}\right)\right) \cdot \left(\sin re \cdot \left(-0.5\right)\right)\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))