Average Error: 43.8 → 0.7
Time: 11.4s
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 r50033 = x;
        double r50034 = exp(r50033);
        double r50035 = -r50033;
        double r50036 = exp(r50035);
        double r50037 = r50034 + r50036;
        double r50038 = 2.0;
        double r50039 = r50037 / r50038;
        double r50040 = y;
        double r50041 = cos(r50040);
        double r50042 = r50039 * r50041;
        double r50043 = r50034 - r50036;
        double r50044 = r50043 / r50038;
        double r50045 = sin(r50040);
        double r50046 = r50044 * r50045;
        double r50047 = /* ERROR: no complex support in C */;
        double r50048 = /* ERROR: no complex support in C */;
        return r50048;
}
↓
double f(double x, double y) {
        double r50049 = 0.3333333333333333;
        double r50050 = x;
        double r50051 = 3.0;
        double r50052 = pow(r50050, r50051);
        double r50053 = 0.016666666666666666;
        double r50054 = 5.0;
        double r50055 = pow(r50050, r50054);
        double r50056 = 2.0;
        double r50057 = r50056 * r50050;
        double r50058 = fma(r50053, r50055, r50057);
        double r50059 = fma(r50049, r50052, r50058);
        double r50060 = 2.0;
        double r50061 = r50059 / r50060;
        double r50062 = y;
        double r50063 = sin(r50062);
        double r50064 = r50061 * r50063;
        return r50064;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020033 +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)))))