math.exp on complex, imaginary part

Average Error: 0.0 → 0.0

Time: 13.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r48346 = re;
        double r48347 = exp(r48346);
        double r48348 = im;
        double r48349 = sin(r48348);
        double r48350 = r48347 * r48349;
        return r48350;
}

↓

double f(double re, double im) {
        double r48351 = re;
        double r48352 = exp(r48351);
        double r48353 = sqrt(r48352);
        double r48354 = sqrt(r48353);
        double r48355 = r48354 * r48354;
        double r48356 = im;
        double r48357 = sin(r48356);
        double r48358 = r48353 * r48357;
        double r48359 = r48355 * r48358;
        return r48359;
}

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 add-sqr-sqrt 0.0

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

7. Applied sqrt-prod 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))