Average Error: 43.6 → 0.8
Time: 30.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r967995 = x;
        double r967996 = exp(r967995);
        double r967997 = -r967995;
        double r967998 = exp(r967997);
        double r967999 = r967996 + r967998;
        double r968000 = 2.0;
        double r968001 = r967999 / r968000;
        double r968002 = y;
        double r968003 = cos(r968002);
        double r968004 = r968001 * r968003;
        double r968005 = r967996 - r967998;
        double r968006 = r968005 / r968000;
        double r968007 = sin(r968002);
        double r968008 = r968006 * r968007;
        double r968009 = /* ERROR: no complex support in C */;
        double r968010 = /* ERROR: no complex support in C */;
        return r968010;
}
↓
double f(double x, double y) {
        double r968011 = x;
        double r968012 = exp(r968011);
        double r968013 = -r968011;
        double r968014 = exp(r968013);
        double r968015 = r968012 + r968014;
        double r968016 = 2.0;
        double r968017 = r968015 / r968016;
        double r968018 = y;
        double r968019 = cos(r968018);
        double r968020 = r968017 * r968019;
        double r968021 = 5.0;
        double r968022 = pow(r968011, r968021);
        double r968023 = 0.016666666666666666;
        double r968024 = 0.3333333333333333;
        double r968025 = r968011 * r968011;
        double r968026 = r968025 * r968011;
        double r968027 = r968024 * r968026;
        double r968028 = fma(r968022, r968023, r968027);
        double r968029 = fma(r968011, r968016, r968028);
        double r968030 = r968029 / r968016;
        double r968031 = sin(r968018);
        double r968032 = r968030 * r968031;
        double r968033 = /* ERROR: no complex support in C */;
        double r968034 = /* ERROR: no complex support in C */;
        return r968034;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019153 +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)))))