Average Error: 43.7 → 0.8
Time: 32.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, 2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}^{5}, \frac{1}{60}, 2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2208944 = x;
        double r2208945 = exp(r2208944);
        double r2208946 = -r2208944;
        double r2208947 = exp(r2208946);
        double r2208948 = r2208945 + r2208947;
        double r2208949 = 2.0;
        double r2208950 = r2208948 / r2208949;
        double r2208951 = y;
        double r2208952 = cos(r2208951);
        double r2208953 = r2208950 * r2208952;
        double r2208954 = r2208945 - r2208947;
        double r2208955 = r2208954 / r2208949;
        double r2208956 = sin(r2208951);
        double r2208957 = r2208955 * r2208956;
        double r2208958 = /* ERROR: no complex support in C */;
        double r2208959 = /* ERROR: no complex support in C */;
        return r2208959;
}
↓
double f(double x, double y) {
        double r2208960 = x;
        double r2208961 = exp(r2208960);
        double r2208962 = -r2208960;
        double r2208963 = exp(r2208962);
        double r2208964 = r2208961 + r2208963;
        double r2208965 = 2.0;
        double r2208966 = r2208964 / r2208965;
        double r2208967 = y;
        double r2208968 = cos(r2208967);
        double r2208969 = r2208966 * r2208968;
        double r2208970 = 5.0;
        double r2208971 = pow(r2208960, r2208970);
        double r2208972 = 0.016666666666666666;
        double r2208973 = r2208965 * r2208960;
        double r2208974 = r2208960 * r2208960;
        double r2208975 = 0.3333333333333333;
        double r2208976 = r2208974 * r2208975;
        double r2208977 = r2208976 * r2208960;
        double r2208978 = r2208973 + r2208977;
        double r2208979 = fma(r2208971, r2208972, r2208978);
        double r2208980 = r2208979 / r2208965;
        double r2208981 = sin(r2208967);
        double r2208982 = r2208980 * r2208981;
        double r2208983 = /* ERROR: no complex support in C */;
        double r2208984 = /* ERROR: no complex support in C */;
        return r2208984;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  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

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

  5. Applied distribute-lft-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 +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)))))