Average Error: 43.3 → 0.7
Time: 30.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(x, 2, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1119115 = x;
        double r1119116 = exp(r1119115);
        double r1119117 = -r1119115;
        double r1119118 = exp(r1119117);
        double r1119119 = r1119116 + r1119118;
        double r1119120 = 2.0;
        double r1119121 = r1119119 / r1119120;
        double r1119122 = y;
        double r1119123 = cos(r1119122);
        double r1119124 = r1119121 * r1119123;
        double r1119125 = r1119116 - r1119118;
        double r1119126 = r1119125 / r1119120;
        double r1119127 = sin(r1119122);
        double r1119128 = r1119126 * r1119127;
        double r1119129 = /* ERROR: no complex support in C */;
        double r1119130 = /* ERROR: no complex support in C */;
        return r1119130;
}
↓
double f(double x, double y) {
        double r1119131 = x;
        double r1119132 = exp(r1119131);
        double r1119133 = -r1119131;
        double r1119134 = exp(r1119133);
        double r1119135 = r1119132 + r1119134;
        double r1119136 = 2.0;
        double r1119137 = r1119135 / r1119136;
        double r1119138 = y;
        double r1119139 = cos(r1119138);
        double r1119140 = r1119137 * r1119139;
        double r1119141 = 5.0;
        double r1119142 = pow(r1119131, r1119141);
        double r1119143 = 0.016666666666666666;
        double r1119144 = 0.3333333333333333;
        double r1119145 = r1119131 * r1119131;
        double r1119146 = r1119131 * r1119145;
        double r1119147 = r1119144 * r1119146;
        double r1119148 = fma(r1119142, r1119143, r1119147);
        double r1119149 = fma(r1119131, r1119136, r1119148);
        double r1119150 = r1119149 / r1119136;
        double r1119151 = sin(r1119138);
        double r1119152 = r1119150 * r1119151;
        double r1119153 = /* ERROR: no complex support in C */;
        double r1119154 = /* ERROR: no complex support in C */;
        return r1119154;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019156 +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)))))