Average Error: 43.1 → 0.8
Time: 54.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1074114 = x;
        double r1074115 = exp(r1074114);
        double r1074116 = -r1074114;
        double r1074117 = exp(r1074116);
        double r1074118 = r1074115 + r1074117;
        double r1074119 = 2.0;
        double r1074120 = r1074118 / r1074119;
        double r1074121 = y;
        double r1074122 = cos(r1074121);
        double r1074123 = r1074120 * r1074122;
        double r1074124 = r1074115 - r1074117;
        double r1074125 = r1074124 / r1074119;
        double r1074126 = sin(r1074121);
        double r1074127 = r1074125 * r1074126;
        double r1074128 = /* ERROR: no complex support in C */;
        double r1074129 = /* ERROR: no complex support in C */;
        return r1074129;
}
↓
double f(double x, double y) {
        double r1074130 = x;
        double r1074131 = exp(r1074130);
        double r1074132 = -r1074130;
        double r1074133 = exp(r1074132);
        double r1074134 = r1074131 + r1074133;
        double r1074135 = 2.0;
        double r1074136 = r1074134 / r1074135;
        double r1074137 = y;
        double r1074138 = cos(r1074137);
        double r1074139 = r1074136 * r1074138;
        double r1074140 = 5.0;
        double r1074141 = pow(r1074130, r1074140);
        double r1074142 = 0.016666666666666666;
        double r1074143 = 0.3333333333333333;
        double r1074144 = r1074130 * r1074130;
        double r1074145 = r1074144 * r1074130;
        double r1074146 = r1074143 * r1074145;
        double r1074147 = fma(r1074141, r1074142, r1074146);
        double r1074148 = fma(r1074130, r1074135, r1074147);
        double r1074149 = r1074148 / r1074135;
        double r1074150 = sin(r1074137);
        double r1074151 = r1074149 * r1074150;
        double r1074152 = /* ERROR: no complex support in C */;
        double r1074153 = /* ERROR: no complex support in C */;
        return r1074153;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.1

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019151 +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)))))