math.cos on complex, imaginary part

Average Error: 43.3 → 0.8

Time: 9.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r198541 = 0.5;
        double r198542 = re;
        double r198543 = sin(r198542);
        double r198544 = r198541 * r198543;
        double r198545 = im;
        double r198546 = -r198545;
        double r198547 = exp(r198546);
        double r198548 = exp(r198545);
        double r198549 = r198547 - r198548;
        double r198550 = r198544 * r198549;
        return r198550;
}

↓

double f(double re, double im) {
        double r198551 = 0.5;
        double r198552 = re;
        double r198553 = sin(r198552);
        double r198554 = -0.3333333333333333;
        double r198555 = im;
        double r198556 = 3.0;
        double r198557 = pow(r198555, r198556);
        double r198558 = r198554 * r198557;
        double r198559 = r198553 * r198558;
        double r198560 = r198551 * r198559;
        double r198561 = r198551 * r198553;
        double r198562 = 0.016666666666666666;
        double r198563 = 5.0;
        double r198564 = pow(r198555, r198563);
        double r198565 = 2.0;
        double r198566 = r198565 * r198555;
        double r198567 = fma(r198562, r198564, r198566);
        double r198568 = -r198567;
        double r198569 = r198561 * r198568;
        double r198570 = r198560 + r198569;
        return r198570;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.3
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.3

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

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

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

5. Applied sub-neg 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Simplified 0.8

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

8. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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