Average Error: 43.7 → 0.8
Time: 34.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, 2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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 r1974021 = x;
        double r1974022 = exp(r1974021);
        double r1974023 = -r1974021;
        double r1974024 = exp(r1974023);
        double r1974025 = r1974022 + r1974024;
        double r1974026 = 2.0;
        double r1974027 = r1974025 / r1974026;
        double r1974028 = y;
        double r1974029 = cos(r1974028);
        double r1974030 = r1974027 * r1974029;
        double r1974031 = r1974022 - r1974024;
        double r1974032 = r1974031 / r1974026;
        double r1974033 = sin(r1974028);
        double r1974034 = r1974032 * r1974033;
        double r1974035 = /* ERROR: no complex support in C */;
        double r1974036 = /* ERROR: no complex support in C */;
        return r1974036;
}
↓
double f(double x, double y) {
        double r1974037 = x;
        double r1974038 = exp(r1974037);
        double r1974039 = -r1974037;
        double r1974040 = exp(r1974039);
        double r1974041 = r1974038 + r1974040;
        double r1974042 = 2.0;
        double r1974043 = r1974041 / r1974042;
        double r1974044 = y;
        double r1974045 = cos(r1974044);
        double r1974046 = r1974043 * r1974045;
        double r1974047 = 5.0;
        double r1974048 = pow(r1974037, r1974047);
        double r1974049 = 0.016666666666666666;
        double r1974050 = r1974042 * r1974037;
        double r1974051 = r1974037 * r1974037;
        double r1974052 = 0.3333333333333333;
        double r1974053 = r1974051 * r1974052;
        double r1974054 = r1974053 * r1974037;
        double r1974055 = r1974050 + r1974054;
        double r1974056 = fma(r1974048, r1974049, r1974055);
        double r1974057 = r1974056 / r1974042;
        double r1974058 = sin(r1974044);
        double r1974059 = r1974057 * r1974058;
        double r1974060 = /* ERROR: no complex support in C */;
        double r1974061 = /* ERROR: no complex support in C */;
        return r1974061;
}

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-rgt-in0.8

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

  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)))))