Euler formula imaginary part (p55)

Average Error: 43.7 → 0.8

Time: 1.6m

Precision: 64

Math

\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

↓

\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]

C

double f(double x, double y) {
        double r5040151 = x;
        double r5040152 = exp(r5040151);
        double r5040153 = -r5040151;
        double r5040154 = exp(r5040153);
        double r5040155 = r5040152 + r5040154;
        double r5040156 = 2.0;
        double r5040157 = r5040155 / r5040156;
        double r5040158 = y;
        double r5040159 = cos(r5040158);
        double r5040160 = r5040157 * r5040159;
        double r5040161 = r5040152 - r5040154;
        double r5040162 = r5040161 / r5040156;
        double r5040163 = sin(r5040158);
        double r5040164 = r5040162 * r5040163;
        double r5040165 = /* ERROR: no complex support in C */;
        double r5040166 = /* ERROR: no complex support in C */;
        return r5040166;
}

↓

double f(double x, double y) {
        double r5040167 = x;
        double r5040168 = exp(r5040167);
        double r5040169 = -r5040167;
        double r5040170 = exp(r5040169);
        double r5040171 = r5040168 + r5040170;
        double r5040172 = 2.0;
        double r5040173 = r5040171 / r5040172;
        double r5040174 = y;
        double r5040175 = cos(r5040174);
        double r5040176 = r5040173 * r5040175;
        double r5040177 = 5.0;
        double r5040178 = pow(r5040167, r5040177);
        double r5040179 = 0.016666666666666666;
        double r5040180 = 0.3333333333333333;
        double r5040181 = r5040167 * r5040167;
        double r5040182 = fma(r5040180, r5040181, r5040172);
        double r5040183 = r5040182 * r5040167;
        double r5040184 = fma(r5040178, r5040179, r5040183);
        double r5040185 = r5040184 / r5040172;
        double r5040186 = sin(r5040174);
        double r5040187 = r5040185 * r5040186;
        double r5040188 = /* ERROR: no complex support in C */;
        double r5040189 = /* ERROR: no complex support in C */;
        return r5040189;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 43.7

   \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

2. Taylor expanded around 0 0.8

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]

3. Simplified 0.8

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}}{2} \cdot \sin y i\right))\]
4. Using strategy rm

5. Applied add-sqr-sqrt 1.6

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \sin y i\right))\]

6. Applied *-un-lft-identity 1.6

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{1 \cdot \mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}}{\sqrt{2} \cdot \sqrt{2}} \cdot \sin y i\right))\]

7. Applied times-frac 1.4

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}}\right)} \cdot \sin y i\right))\]

8. Applied associate-*l* 1.4

   \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}} \cdot \sin y\right)} i\right))\]
9. Using strategy rm

10. Applied pow1 1.4

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{1}{\sqrt{2}} \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}} \cdot \color{blue}{{\left(\sin y\right)}^{1}}\right) i\right))\]

11. Applied pow1 1.4

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{1}{\sqrt{2}} \cdot \left(\color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}}\right)}^{1}} \cdot {\left(\sin y\right)}^{1}\right) i\right))\]

12. Applied pow-prod-down 1.4

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{1}{\sqrt{2}} \cdot \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}} \cdot \sin y\right)}^{1}} i\right))\]

13. Applied pow1 1.4

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{{\left(\frac{1}{\sqrt{2}}\right)}^{1}} \cdot {\left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}} \cdot \sin y\right)}^{1} i\right))\]

14. Applied pow-prod-down 1.4

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{{\left(\frac{1}{\sqrt{2}} \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\sqrt{2}} \cdot \sin y\right)\right)}^{1}} i\right))\]

15. Simplified 0.9

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + {\color{blue}{\left(\frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{\frac{2}{\sin y}}\right)}}^{1} i\right))\]
16. Using strategy rm

17. Applied associate-/r/ 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + {\color{blue}{\left(\frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(x \cdot \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right)\right)\right)}{2} \cdot \sin y\right)}}^{1} i\right))\]

18. Final simplification 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))