math.cos on complex, imaginary part

Average Error: 43.8 → 0.9

Time: 55.0s

Precision: 64

Math

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

↓

\[-\mathsf{fma}\left(\sin re, {im}^{5} \cdot 0.008333333333333333217685101601546193705872, \mathsf{fma}\left(1, im \cdot \sin re, \left(0.1666666666666666574148081281236954964697 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \sin re\right)\right)\right)\]

C

double f(double re, double im) {
        double r7672790 = 0.5;
        double r7672791 = re;
        double r7672792 = sin(r7672791);
        double r7672793 = r7672790 * r7672792;
        double r7672794 = im;
        double r7672795 = -r7672794;
        double r7672796 = exp(r7672795);
        double r7672797 = exp(r7672794);
        double r7672798 = r7672796 - r7672797;
        double r7672799 = r7672793 * r7672798;
        return r7672799;
}

↓

double f(double re, double im) {
        double r7672800 = re;
        double r7672801 = sin(r7672800);
        double r7672802 = im;
        double r7672803 = 5.0;
        double r7672804 = pow(r7672802, r7672803);
        double r7672805 = 0.008333333333333333;
        double r7672806 = r7672804 * r7672805;
        double r7672807 = 1.0;
        double r7672808 = r7672802 * r7672801;
        double r7672809 = 0.16666666666666666;
        double r7672810 = r7672802 * r7672802;
        double r7672811 = r7672809 * r7672810;
        double r7672812 = r7672811 * r7672808;
        double r7672813 = fma(r7672807, r7672808, r7672812);
        double r7672814 = fma(r7672801, r7672806, r7672813);
        double r7672815 = -r7672814;
        return r7672815;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.8
Target	0.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.8

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

2. Taylor expanded around 0 0.9

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

3. Simplified 0.9

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

4. Taylor expanded around inf 0.9

   \[\leadsto \color{blue}{-\left(0.1666666666666666574148081281236954964697 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333217685101601546193705872 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

5. Simplified 0.9

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sin re, {im}^{5} \cdot 0.008333333333333333217685101601546193705872, \mathsf{fma}\left(1, im \cdot \sin re, \left(0.1666666666666666574148081281236954964697 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \sin re\right)\right)\right)}\]

6. Final simplification 0.9

   \[\leadsto -\mathsf{fma}\left(\sin re, {im}^{5} \cdot 0.008333333333333333217685101601546193705872, \mathsf{fma}\left(1, im \cdot \sin re, \left(0.1666666666666666574148081281236954964697 \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \sin re\right)\right)\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))