math.cos on complex, imaginary part

Average Error: 43.4 → 0.8

Time: 52.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r37057015 = 0.5;
        double r37057016 = re;
        double r37057017 = sin(r37057016);
        double r37057018 = r37057015 * r37057017;
        double r37057019 = im;
        double r37057020 = -r37057019;
        double r37057021 = exp(r37057020);
        double r37057022 = exp(r37057019);
        double r37057023 = r37057021 - r37057022;
        double r37057024 = r37057018 * r37057023;
        return r37057024;
}

↓

double f(double re, double im) {
        double r37057025 = im;
        double r37057026 = 5.0;
        double r37057027 = pow(r37057025, r37057026);
        double r37057028 = -0.016666666666666666;
        double r37057029 = -2.0;
        double r37057030 = r37057025 * r37057029;
        double r37057031 = -0.3333333333333333;
        double r37057032 = r37057025 * r37057031;
        double r37057033 = r37057025 * r37057032;
        double r37057034 = r37057025 * r37057033;
        double r37057035 = r37057030 + r37057034;
        double r37057036 = fma(r37057027, r37057028, r37057035);
        double r37057037 = 0.5;
        double r37057038 = re;
        double r37057039 = sin(r37057038);
        double r37057040 = r37057037 * r37057039;
        double r37057041 = r37057036 * r37057040;
        return r37057041;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.4
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 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

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

5. Applied sub-neg 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Simplified 0.8

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

8. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(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))))