Average Error: 42.9 → 0.4
Time: 31.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.015295351353249335:\\ \;\;\;\;0.5 \cdot \left(\sin y \cdot \left(e^{x} - e^{-x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\\ \end{array}\]
\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 -0.015295351353249335:\\
\;\;\;\;0.5 \cdot \left(\sin y \cdot \left(e^{x} - e^{-x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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\\

\end{array}
double f(double x, double y) {
        double r53938 = x;
        double r53939 = exp(r53938);
        double r53940 = -r53938;
        double r53941 = exp(r53940);
        double r53942 = r53939 + r53941;
        double r53943 = 2.0;
        double r53944 = r53942 / r53943;
        double r53945 = y;
        double r53946 = cos(r53945);
        double r53947 = r53944 * r53946;
        double r53948 = r53939 - r53941;
        double r53949 = r53948 / r53943;
        double r53950 = sin(r53945);
        double r53951 = r53949 * r53950;
        double r53952 = /* ERROR: no complex support in C */;
        double r53953 = /* ERROR: no complex support in C */;
        return r53953;
}
↓
double f(double x, double y) {
        double r53954 = x;
        double r53955 = -0.015295351353249335;
        bool r53956 = r53954 <= r53955;
        double r53957 = 0.5;
        double r53958 = y;
        double r53959 = sin(r53958);
        double r53960 = exp(r53954);
        double r53961 = -r53954;
        double r53962 = exp(r53961);
        double r53963 = r53960 - r53962;
        double r53964 = r53959 * r53963;
        double r53965 = r53957 * r53964;
        double r53966 = 0.3333333333333333;
        double r53967 = 3.0;
        double r53968 = pow(r53954, r53967);
        double r53969 = 0.016666666666666666;
        double r53970 = 5.0;
        double r53971 = pow(r53954, r53970);
        double r53972 = 2.0;
        double r53973 = r53972 * r53954;
        double r53974 = fma(r53969, r53971, r53973);
        double r53975 = fma(r53966, r53968, r53974);
        double r53976 = 2.0;
        double r53977 = r53975 / r53976;
        double r53978 = r53977 * r53959;
        double r53979 = r53956 ? r53965 : r53978;
        return r53979;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.015295351353249335

    1. Initial program 1.6

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

    2. Simplified1.6

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

    3. Taylor expanded around inf 1.6

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin y \cdot \left(e^{x} - e^{-x}\right)\right)}\]

    if -0.015295351353249335 < x

    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 0.4

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

    4. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.015295351353249335:\\ \;\;\;\;0.5 \cdot \left(\sin y \cdot \left(e^{x} - e^{-x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))