Average Error: 33.6 → 3.5
Time: 8.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\]
e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
↓
e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14208 = x_re;
        double r14209 = r14208 * r14208;
        double r14210 = x_im;
        double r14211 = r14210 * r14210;
        double r14212 = r14209 + r14211;
        double r14213 = sqrt(r14212);
        double r14214 = log(r14213);
        double r14215 = y_re;
        double r14216 = r14214 * r14215;
        double r14217 = atan2(r14210, r14208);
        double r14218 = y_im;
        double r14219 = r14217 * r14218;
        double r14220 = r14216 - r14219;
        double r14221 = exp(r14220);
        double r14222 = r14214 * r14218;
        double r14223 = r14217 * r14215;
        double r14224 = r14222 + r14223;
        double r14225 = sin(r14224);
        double r14226 = r14221 * r14225;
        return r14226;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14227 = x_re;
        double r14228 = x_im;
        double r14229 = hypot(r14227, r14228);
        double r14230 = log(r14229);
        double r14231 = exp(r14230);
        double r14232 = log(r14231);
        double r14233 = y_re;
        double r14234 = r14232 * r14233;
        double r14235 = atan2(r14228, r14227);
        double r14236 = y_im;
        double r14237 = r14235 * r14236;
        double r14238 = r14234 - r14237;
        double r14239 = exp(r14238);
        double r14240 = r14230 * r14236;
        double r14241 = r14235 * r14233;
        double r14242 = r14240 + r14241;
        double r14243 = sin(r14242);
        double r14244 = log1p(r14243);
        double r14245 = expm1(r14244);
        double r14246 = r14239 * r14245;
        return r14246;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 33.6

    \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  2. Using strategy rm

  3. Applied hypot-def19.2

    \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  4. Using strategy rm

  5. Applied add-exp-log19.2

    \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

  6. Simplified3.5

    \[\leadsto e^{\log \left(e^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
  7. Using strategy rm

  8. Applied expm1-log1p-u3.5

    \[\leadsto e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}\]

  9. Final simplification3.5

    \[\leadsto e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))