math.cos on complex, imaginary part

Average Error: 44.0 → 0.8

Time: 44.1s

Precision: 64

Math

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

↓

\[-\left(\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333 + im \cdot 1.0\right) \cdot \sin re + \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right)\right)\]

C

double f(double re, double im) {
        double r9342201 = 0.5;
        double r9342202 = re;
        double r9342203 = sin(r9342202);
        double r9342204 = r9342201 * r9342203;
        double r9342205 = im;
        double r9342206 = -r9342205;
        double r9342207 = exp(r9342206);
        double r9342208 = exp(r9342205);
        double r9342209 = r9342207 - r9342208;
        double r9342210 = r9342204 * r9342209;
        return r9342210;
}

↓

double f(double re, double im) {
        double r9342211 = im;
        double r9342212 = r9342211 * r9342211;
        double r9342213 = r9342211 * r9342212;
        double r9342214 = r9342213 * r9342212;
        double r9342215 = 0.008333333333333333;
        double r9342216 = r9342214 * r9342215;
        double r9342217 = 1.0;
        double r9342218 = r9342211 * r9342217;
        double r9342219 = r9342216 + r9342218;
        double r9342220 = re;
        double r9342221 = sin(r9342220);
        double r9342222 = r9342219 * r9342221;
        double r9342223 = 0.16666666666666666;
        double r9342224 = r9342213 * r9342223;
        double r9342225 = r9342221 * r9342224;
        double r9342226 = r9342222 + r9342225;
        double r9342227 = -r9342226;
        return r9342227;
}

Error

Bits error versus re

Bits error versus im

Target

Original	44.0
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 44.0

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

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

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

4. Taylor expanded around inf 0.8

   \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

5. Simplified 0.8

   \[\leadsto \color{blue}{-\left(\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot \sin re + \sin re \cdot \left(im \cdot 1.0 + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333\right)\right)}\]

6. Final simplification 0.8

   \[\leadsto -\left(\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333 + im \cdot 1.0\right) \cdot \sin re + \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right)\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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