Average Error: 43.0 → 0.8
Time: 55.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1729961 = x;
        double r1729962 = exp(r1729961);
        double r1729963 = -r1729961;
        double r1729964 = exp(r1729963);
        double r1729965 = r1729962 + r1729964;
        double r1729966 = 2.0;
        double r1729967 = r1729965 / r1729966;
        double r1729968 = y;
        double r1729969 = cos(r1729968);
        double r1729970 = r1729967 * r1729969;
        double r1729971 = r1729962 - r1729964;
        double r1729972 = r1729971 / r1729966;
        double r1729973 = sin(r1729968);
        double r1729974 = r1729972 * r1729973;
        double r1729975 = /* ERROR: no complex support in C */;
        double r1729976 = /* ERROR: no complex support in C */;
        return r1729976;
}
↓
double f(double x, double y) {
        double r1729977 = x;
        double r1729978 = exp(r1729977);
        double r1729979 = -r1729977;
        double r1729980 = exp(r1729979);
        double r1729981 = r1729978 + r1729980;
        double r1729982 = 2.0;
        double r1729983 = r1729981 / r1729982;
        double r1729984 = y;
        double r1729985 = cos(r1729984);
        double r1729986 = r1729983 * r1729985;
        double r1729987 = 5.0;
        double r1729988 = pow(r1729977, r1729987);
        double r1729989 = 0.016666666666666666;
        double r1729990 = 0.3333333333333333;
        double r1729991 = r1729977 * r1729977;
        double r1729992 = r1729991 * r1729977;
        double r1729993 = r1729990 * r1729992;
        double r1729994 = fma(r1729988, r1729989, r1729993);
        double r1729995 = fma(r1729977, r1729982, r1729994);
        double r1729996 = r1729995 / r1729982;
        double r1729997 = sin(r1729984);
        double r1729998 = r1729996 * r1729997;
        double r1729999 = /* ERROR: no complex support in C */;
        double r1730000 = /* ERROR: no complex support in C */;
        return r1730000;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.0

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019130 +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)))))