math.cos on complex, imaginary part

Average Error: 43.6 → 0.8

Time: 8.9s

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 r271567 = 0.5;
        double r271568 = re;
        double r271569 = sin(r271568);
        double r271570 = r271567 * r271569;
        double r271571 = im;
        double r271572 = -r271571;
        double r271573 = exp(r271572);
        double r271574 = exp(r271571);
        double r271575 = r271573 - r271574;
        double r271576 = r271570 * r271575;
        return r271576;
}

↓

double f(double re, double im) {
        double r271577 = 0.5;
        double r271578 = re;
        double r271579 = sin(r271578);
        double r271580 = -0.3333333333333333;
        double r271581 = im;
        double r271582 = 3.0;
        double r271583 = pow(r271581, r271582);
        double r271584 = r271580 * r271583;
        double r271585 = r271579 * r271584;
        double r271586 = r271577 * r271585;
        double r271587 = r271577 * r271579;
        double r271588 = 0.016666666666666666;
        double r271589 = 5.0;
        double r271590 = pow(r271581, r271589);
        double r271591 = 2.0;
        double r271592 = r271591 * r271581;
        double r271593 = fma(r271588, r271590, r271592);
        double r271594 = -r271593;
        double r271595 = r271587 * r271594;
        double r271596 = r271586 + r271595;
        return r271596;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.6
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 43.6

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

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 2020018 +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))))