math.cos on complex, imaginary part

Average Error: 43.6 → 0.7

Time: 32.3s

Precision: 64

Math

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

↓

\[-\mathsf{fma}\left(\left(0.16666666666666666 \cdot \sin re\right), \left(im \cdot \left(im \cdot im\right)\right), \left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \sin re\right), \left({im}^{5}\right), \left(1.0 \cdot \left(im \cdot \sin re\right)\right)\right)\right)\right)\]

C

double f(double re, double im) {
        double r3443389 = 0.5;
        double r3443390 = re;
        double r3443391 = sin(r3443390);
        double r3443392 = r3443389 * r3443391;
        double r3443393 = im;
        double r3443394 = -r3443393;
        double r3443395 = exp(r3443394);
        double r3443396 = exp(r3443393);
        double r3443397 = r3443395 - r3443396;
        double r3443398 = r3443392 * r3443397;
        return r3443398;
}

↓

double f(double re, double im) {
        double r3443399 = 0.16666666666666666;
        double r3443400 = re;
        double r3443401 = sin(r3443400);
        double r3443402 = r3443399 * r3443401;
        double r3443403 = im;
        double r3443404 = r3443403 * r3443403;
        double r3443405 = r3443403 * r3443404;
        double r3443406 = 0.008333333333333333;
        double r3443407 = r3443406 * r3443401;
        double r3443408 = 5.0;
        double r3443409 = pow(r3443403, r3443408);
        double r3443410 = 1.0;
        double r3443411 = r3443403 * r3443401;
        double r3443412 = r3443410 * r3443411;
        double r3443413 = fma(r3443407, r3443409, r3443412);
        double r3443414 = fma(r3443402, r3443405, r3443413);
        double r3443415 = -r3443414;
        return r3443415;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.6
Target	0.3
Herbie	0.7

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

Derivation

1. Initial program 43.6

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

2. Taylor expanded around 0 0.7

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

3. Simplified 0.7

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

4. Taylor expanded around -inf 0.7

   \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

5. Simplified 0.7

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\left(0.16666666666666666 \cdot \sin re\right), \left(im \cdot \left(im \cdot im\right)\right), \left(\mathsf{fma}\left(\left(\sin re \cdot 0.008333333333333333\right), \left({im}^{5}\right), \left(1.0 \cdot \left(im \cdot \sin re\right)\right)\right)\right)\right)}\]

6. Final simplification 0.7

   \[\leadsto -\mathsf{fma}\left(\left(0.16666666666666666 \cdot \sin re\right), \left(im \cdot \left(im \cdot im\right)\right), \left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \sin re\right), \left({im}^{5}\right), \left(1.0 \cdot \left(im \cdot \sin re\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019132 +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))))