math.exp on complex, imaginary part

Average Error: 0.0 → 0.0

Time: 15.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[e^{re} \cdot \sin im\]

↓

\[\sin im \cdot e^{re}\]

C

double f(double re, double im) {
        double r45392 = re;
        double r45393 = exp(r45392);
        double r45394 = im;
        double r45395 = sin(r45394);
        double r45396 = r45393 * r45395;
        return r45396;
}

↓

double f(double re, double im) {
        double r45397 = im;
        double r45398 = sin(r45397);
        double r45399 = re;
        double r45400 = exp(r45399);
        double r45401 = r45398 * r45400;
        return r45401;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.0

   \[e^{re} \cdot \sin im\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.0

   \[\leadsto \color{blue}{\left(\sqrt{e^{re}} \cdot \sqrt{e^{re}}\right)} \cdot \sin im\]

4. Applied associate-*l* 0.0

   \[\leadsto \color{blue}{\sqrt{e^{re}} \cdot \left(\sqrt{e^{re}} \cdot \sin im\right)}\]
5. Using strategy rm

6. Applied pow1 0.0

   \[\leadsto \sqrt{e^{re}} \cdot \left(\sqrt{e^{re}} \cdot \color{blue}{{\left(\sin im\right)}^{1}}\right)\]

7. Applied pow1 0.0

   \[\leadsto \sqrt{e^{re}} \cdot \left(\color{blue}{{\left(\sqrt{e^{re}}\right)}^{1}} \cdot {\left(\sin im\right)}^{1}\right)\]

8. Applied pow-prod-down 0.0

   \[\leadsto \sqrt{e^{re}} \cdot \color{blue}{{\left(\sqrt{e^{re}} \cdot \sin im\right)}^{1}}\]

9. Applied pow1 0.0

   \[\leadsto \color{blue}{{\left(\sqrt{e^{re}}\right)}^{1}} \cdot {\left(\sqrt{e^{re}} \cdot \sin im\right)}^{1}\]

10. Applied pow-prod-down 0.0

    \[\leadsto \color{blue}{{\left(\sqrt{e^{re}} \cdot \left(\sqrt{e^{re}} \cdot \sin im\right)\right)}^{1}}\]

11. Simplified 0.0

    \[\leadsto {\color{blue}{\left(e^{re} \cdot \sin im\right)}}^{1}\]

12. Final simplification 0.0

    \[\leadsto \sin im \cdot e^{re}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))