Average Error: 42.9 → 0.8
Time: 1.7m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{60}, \left({x}^{5}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right), 2\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r6552000 = x;
        double r6552001 = exp(r6552000);
        double r6552002 = -r6552000;
        double r6552003 = exp(r6552002);
        double r6552004 = r6552001 + r6552003;
        double r6552005 = 2.0;
        double r6552006 = r6552004 / r6552005;
        double r6552007 = y;
        double r6552008 = cos(r6552007);
        double r6552009 = r6552006 * r6552008;
        double r6552010 = r6552001 - r6552003;
        double r6552011 = r6552010 / r6552005;
        double r6552012 = sin(r6552007);
        double r6552013 = r6552011 * r6552012;
        double r6552014 = /* ERROR: no complex support in C */;
        double r6552015 = /* ERROR: no complex support in C */;
        return r6552015;
}
↓
double f(double x, double y) {
        double r6552016 = x;
        double r6552017 = exp(r6552016);
        double r6552018 = -r6552016;
        double r6552019 = exp(r6552018);
        double r6552020 = r6552017 + r6552019;
        double r6552021 = 2.0;
        double r6552022 = r6552020 / r6552021;
        double r6552023 = y;
        double r6552024 = cos(r6552023);
        double r6552025 = r6552022 * r6552024;
        double r6552026 = 0.016666666666666666;
        double r6552027 = 5.0;
        double r6552028 = pow(r6552016, r6552027);
        double r6552029 = 0.3333333333333333;
        double r6552030 = r6552016 * r6552016;
        double r6552031 = fma(r6552029, r6552030, r6552021);
        double r6552032 = r6552031 * r6552016;
        double r6552033 = fma(r6552026, r6552028, r6552032);
        double r6552034 = r6552033 / r6552021;
        double r6552035 = sin(r6552023);
        double r6552036 = r6552034 * r6552035;
        double r6552037 = /* ERROR: no complex support in C */;
        double r6552038 = /* ERROR: no complex support in C */;
        return r6552038;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 42.9

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

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019125 +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)))))