Euler formula imaginary part (p55)

Average Error: 43.7 → 0.9

Time: 1.2m

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r4119597 = x;
        double r4119598 = exp(r4119597);
        double r4119599 = -r4119597;
        double r4119600 = exp(r4119599);
        double r4119601 = r4119598 + r4119600;
        double r4119602 = 2.0;
        double r4119603 = r4119601 / r4119602;
        double r4119604 = y;
        double r4119605 = cos(r4119604);
        double r4119606 = r4119603 * r4119605;
        double r4119607 = r4119598 - r4119600;
        double r4119608 = r4119607 / r4119602;
        double r4119609 = sin(r4119604);
        double r4119610 = r4119608 * r4119609;
        double r4119611 = /* ERROR: no complex support in C */;
        double r4119612 = /* ERROR: no complex support in C */;
        return r4119612;
}

↓

double f(double x, double y) {
        double r4119613 = x;
        double r4119614 = exp(r4119613);
        double r4119615 = -r4119613;
        double r4119616 = exp(r4119615);
        double r4119617 = r4119614 + r4119616;
        double r4119618 = 2.0;
        double r4119619 = r4119617 / r4119618;
        double r4119620 = y;
        double r4119621 = cos(r4119620);
        double r4119622 = r4119619 * r4119621;
        double r4119623 = 0.3333333333333333;
        double r4119624 = r4119613 * r4119623;
        double r4119625 = r4119624 * r4119613;
        double r4119626 = r4119625 + r4119618;
        double r4119627 = r4119613 * r4119626;
        double r4119628 = 5.0;
        double r4119629 = pow(r4119613, r4119628);
        double r4119630 = 0.016666666666666666;
        double r4119631 = r4119629 * r4119630;
        double r4119632 = r4119627 + r4119631;
        double r4119633 = sin(r4119620);
        double r4119634 = r4119618 / r4119633;
        double r4119635 = r4119632 / r4119634;
        double r4119636 = /* ERROR: no complex support in C */;
        double r4119637 = /* ERROR: no complex support in C */;
        return r4119637;
}

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

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

5. Applied add-sqr-sqrt 1.6

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

6. Applied *-un-lft-identity 1.6

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

7. Applied times-frac 1.4

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

8. Applied associate-*l* 1.4

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

10. Applied pow1 1.4

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

11. Applied pow1 1.4

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

12. Applied pow-prod-down 1.4

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

13. Applied pow1 1.4

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

14. Applied pow-prod-down 1.4

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

15. Simplified 0.9

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

16. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019121 
(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)))))