math.cos on complex, imaginary part

Average Error: 43.4 → 0.8

Time: 32.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r219086 = 0.5;
        double r219087 = re;
        double r219088 = sin(r219087);
        double r219089 = r219086 * r219088;
        double r219090 = im;
        double r219091 = -r219090;
        double r219092 = exp(r219091);
        double r219093 = exp(r219090);
        double r219094 = r219092 - r219093;
        double r219095 = r219089 * r219094;
        return r219095;
}

↓

double f(double re, double im) {
        double r219096 = 0.5;
        double r219097 = re;
        double r219098 = sin(r219097);
        double r219099 = r219096 * r219098;
        double r219100 = -0.3333333333333333;
        double r219101 = im;
        double r219102 = 3.0;
        double r219103 = pow(r219101, r219102);
        double r219104 = r219100 * r219103;
        double r219105 = r219099 * r219104;
        double r219106 = -2.0;
        double r219107 = 5.0;
        double r219108 = pow(r219101, r219107);
        double r219109 = -0.016666666666666666;
        double r219110 = r219108 * r219109;
        double r219111 = fma(r219101, r219106, r219110);
        double r219112 = r219099 * r219111;
        double r219113 = r219105 + r219112;
        return r219113;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.4
Target	0.2
Herbie	0.8

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

Derivation

1. Initial program 43.4

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

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

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

5. Applied fma-udef 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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