Average Error: 43.2 → 0.5
Time: 35.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01288229977934218090096507580710749607533:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \left(\sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\right) \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} i\right))\\ \mathbf{else}:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\\ \end{array}\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01288229977934218090096507580710749607533:\\
\;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \left(\sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\right) \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} i\right))\\

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

\end{array}
double f(double x, double y) {
        double r56046 = x;
        double r56047 = exp(r56046);
        double r56048 = -r56046;
        double r56049 = exp(r56048);
        double r56050 = r56047 + r56049;
        double r56051 = 2.0;
        double r56052 = r56050 / r56051;
        double r56053 = y;
        double r56054 = cos(r56053);
        double r56055 = r56052 * r56054;
        double r56056 = r56047 - r56049;
        double r56057 = r56056 / r56051;
        double r56058 = sin(r56053);
        double r56059 = r56057 * r56058;
        double r56060 = /* ERROR: no complex support in C */;
        double r56061 = /* ERROR: no complex support in C */;
        return r56061;
}
↓
double f(double x, double y) {
        double r56062 = x;
        double r56063 = -0.012882299779342181;
        bool r56064 = r56062 <= r56063;
        double r56065 = exp(r56062);
        double r56066 = -r56062;
        double r56067 = exp(r56066);
        double r56068 = r56065 + r56067;
        double r56069 = 2.0;
        double r56070 = r56068 / r56069;
        double r56071 = y;
        double r56072 = cos(r56071);
        double r56073 = r56070 * r56072;
        double r56074 = r56065 - r56067;
        double r56075 = r56074 / r56069;
        double r56076 = sin(r56071);
        double r56077 = r56075 * r56076;
        double r56078 = cbrt(r56077);
        double r56079 = r56078 * r56078;
        double r56080 = r56079 * r56078;
        double r56081 = /* ERROR: no complex support in C */;
        double r56082 = /* ERROR: no complex support in C */;
        double r56083 = 0.3333333333333333;
        double r56084 = 3.0;
        double r56085 = pow(r56062, r56084);
        double r56086 = r56083 * r56085;
        double r56087 = 0.016666666666666666;
        double r56088 = 5.0;
        double r56089 = pow(r56062, r56088);
        double r56090 = r56087 * r56089;
        double r56091 = 2.0;
        double r56092 = r56091 * r56062;
        double r56093 = r56090 + r56092;
        double r56094 = r56086 + r56093;
        double r56095 = r56094 / r56069;
        double r56096 = r56095 * r56076;
        double r56097 = /* ERROR: no complex support in C */;
        double r56098 = /* ERROR: no complex support in C */;
        double r56099 = r56064 ? r56082 : r56098;
        return r56099;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.012882299779342181

    1. Initial program 2.8

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

    3. Applied add-cube-cbrt3.5

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

    if -0.012882299779342181 < x

    1. Initial program 43.5

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

    2. Taylor expanded around 0 0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01288229977934218090096507580710749607533:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \left(\sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\right) \cdot \sqrt[3]{\frac{e^{x} - e^{-x}}{2} \cdot \sin y} i\right))\\ \mathbf{else}:\\ \;\;\;\;\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y i\right))\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))