Average Error: 43.2 → 0.8
Time: 32.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r42027 = x;
        double r42028 = exp(r42027);
        double r42029 = -r42027;
        double r42030 = exp(r42029);
        double r42031 = r42028 + r42030;
        double r42032 = 2.0;
        double r42033 = r42031 / r42032;
        double r42034 = y;
        double r42035 = cos(r42034);
        double r42036 = r42033 * r42035;
        double r42037 = r42028 - r42030;
        double r42038 = r42037 / r42032;
        double r42039 = sin(r42034);
        double r42040 = r42038 * r42039;
        double r42041 = /* ERROR: no complex support in C */;
        double r42042 = /* ERROR: no complex support in C */;
        return r42042;
}
↓
double f(double x, double y) {
        double r42043 = 0.3333333333333333;
        double r42044 = x;
        double r42045 = 3.0;
        double r42046 = pow(r42044, r42045);
        double r42047 = 0.016666666666666666;
        double r42048 = 5.0;
        double r42049 = pow(r42044, r42048);
        double r42050 = 2.0;
        double r42051 = r42050 * r42044;
        double r42052 = fma(r42047, r42049, r42051);
        double r42053 = fma(r42043, r42046, r42052);
        double r42054 = 2.0;
        double r42055 = r42053 / r42054;
        double r42056 = y;
        double r42057 = sin(r42056);
        double r42058 = r42055 * r42057;
        return r42058;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))