math.sin on complex, imaginary part

Average Error: 57.8 → 0.5

Time: 9.1s

Precision: 64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right) \le -0.04842324487903843:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im))) <= -0.04842324487903843)) {
		VAR = ((0.5 * cos(re)) * (exp((0.0 - im)) - exp(im)));
	} else {
		VAR = ((0.5 * cos(re)) * -(((0.3333333333333333 * pow(im, 3.0)) + (0.016666666666666666 * pow(im, 5.0))) + (2.0 * im)));
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	57.8
Target	0.3
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))) < -0.04842324487903843

   1. Initial program 0.5

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

   if -0.04842324487903843 < (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))

   1. Initial program 58.3

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

   2. Taylor expanded around 0 0.5

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

   4. Applied associate-+r+ 0.5

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right) \le -0.04842324487903843:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))