math.sin on complex, imaginary part

Average Error: 57.9 → 0.4

Time: 9.9s

Precision: binary64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \le 0.00036555420655970394:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left({im}^{3} \cdot -0.3333333333333333 + \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) (((double) (0.5 * ((double) cos(re)))) * ((double) (((double) exp(((double) (0.0 - im)))) - ((double) exp(im))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((((double) (((double) exp(((double) (0.0 - im)))) - ((double) exp(im)))) <= 0.00036555420655970394)) {
		VAR = ((double) (((double) (0.5 * ((double) cos(re)))) * ((double) (((double) (((double) pow(im, 3.0)) * -0.3333333333333333)) + ((double) (((double) (im * -2.0)) + ((double) (((double) pow(im, 5.0)) * -0.016666666666666666))))))));
	} else {
		VAR = ((double) (((double) (0.5 * ((double) cos(re)))) * ((double) (((double) exp(((double) -(im)))) - ((double) exp(im))))));
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	57.9
Target	0.3
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (exp (- 0.0 im)) (exp im)) < 3.655542065597e-4

   1. Initial program 58.5

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

   2. Taylor expanded around 0 0.4

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(0.3333333333333333 \cdot {im}^{3} + \left(0.016666666666666666 \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

   3. Simplified 0.4

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left({im}^{3} \cdot -0.3333333333333333 + \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\right)}\]

   if 3.655542065597e-4 < (- (exp (- 0.0 im)) (exp im))

   1. Initial program 2.0

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

   2. Taylor expanded around inf 2.0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{-im} - e^{im}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \le 0.00036555420655970394:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left({im}^{3} \cdot -0.3333333333333333 + \left(im \cdot -2 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))