math.sin on complex, real part

Average Error: 0.0 → 0.3

Time: 20.4s

Precision: 64

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

↓

\[\frac{\sin re \cdot \left(\left(e^{im} \cdot 0.5\right) \cdot \left(\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right)\right) + \sqrt[3]{\left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right)\right) \cdot \left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right)\right)} \cdot \frac{0.5}{e^{im}}\right)}{\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right) + \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}} - \frac{0.5}{e^{im}} \cdot \left(e^{im} \cdot 0.5\right)\right)}\]

C

double f(double re, double im) {
        double r494696 = 0.5;
        double r494697 = re;
        double r494698 = sin(r494697);
        double r494699 = r494696 * r494698;
        double r494700 = 0.0;
        double r494701 = im;
        double r494702 = r494700 - r494701;
        double r494703 = exp(r494702);
        double r494704 = exp(r494701);
        double r494705 = r494703 + r494704;
        double r494706 = r494699 * r494705;
        return r494706;
}

↓

double f(double re, double im) {
        double r494707 = re;
        double r494708 = sin(r494707);
        double r494709 = im;
        double r494710 = exp(r494709);
        double r494711 = 0.5;
        double r494712 = r494710 * r494711;
        double r494713 = r494712 * r494712;
        double r494714 = r494712 * r494713;
        double r494715 = r494711 / r494710;
        double r494716 = r494715 * r494715;
        double r494717 = r494715 * r494716;
        double r494718 = r494717 * r494717;
        double r494719 = cbrt(r494718);
        double r494720 = r494719 * r494715;
        double r494721 = r494714 + r494720;
        double r494722 = r494708 * r494721;
        double r494723 = r494715 * r494712;
        double r494724 = r494716 - r494723;
        double r494725 = r494713 + r494724;
        double r494726 = r494722 / r494725;
        return r494726;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.0

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

2. Simplified 0.0

   \[\leadsto \color{blue}{\left(e^{im} \cdot 0.5 + \frac{0.5}{e^{im}}\right) \cdot \sin re}\]
3. Using strategy rm

4. Applied flip3-+ 0.2

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

5. Applied associate-*l/ 0.2

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

6. Simplified 0.2

   \[\leadsto \frac{\color{blue}{\sin re \cdot \left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right) + \left(\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right)\right) \cdot \left(e^{im} \cdot 0.5\right)\right)}}{\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right) + \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}} - \left(e^{im} \cdot 0.5\right) \cdot \frac{0.5}{e^{im}}\right)}\]
7. Using strategy rm

8. Applied add-cbrt-cube 0.2

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

9. Applied add-cbrt-cube 1.2

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

10. Applied cbrt-unprod 0.3

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

11. Final simplification 0.3

    \[\leadsto \frac{\sin re \cdot \left(\left(e^{im} \cdot 0.5\right) \cdot \left(\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right)\right) + \sqrt[3]{\left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right)\right) \cdot \left(\frac{0.5}{e^{im}} \cdot \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}}\right)\right)} \cdot \frac{0.5}{e^{im}}\right)}{\left(e^{im} \cdot 0.5\right) \cdot \left(e^{im} \cdot 0.5\right) + \left(\frac{0.5}{e^{im}} \cdot \frac{0.5}{e^{im}} - \frac{0.5}{e^{im}} \cdot \left(e^{im} \cdot 0.5\right)\right)}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))