Average Error: 43.2 → 0.8
Time: 35.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49155 = x;
        double r49156 = exp(r49155);
        double r49157 = -r49155;
        double r49158 = exp(r49157);
        double r49159 = r49156 + r49158;
        double r49160 = 2.0;
        double r49161 = r49159 / r49160;
        double r49162 = y;
        double r49163 = cos(r49162);
        double r49164 = r49161 * r49163;
        double r49165 = r49156 - r49158;
        double r49166 = r49165 / r49160;
        double r49167 = sin(r49162);
        double r49168 = r49166 * r49167;
        double r49169 = /* ERROR: no complex support in C */;
        double r49170 = /* ERROR: no complex support in C */;
        return r49170;
}
↓
double f(double x, double y) {
        double r49171 = x;
        double r49172 = exp(r49171);
        double r49173 = -r49171;
        double r49174 = exp(r49173);
        double r49175 = r49172 + r49174;
        double r49176 = 2.0;
        double r49177 = r49175 / r49176;
        double r49178 = y;
        double r49179 = cos(r49178);
        double r49180 = r49177 * r49179;
        double r49181 = 0.3333333333333333;
        double r49182 = 3.0;
        double r49183 = pow(r49171, r49182);
        double r49184 = 0.016666666666666666;
        double r49185 = 5.0;
        double r49186 = pow(r49171, r49185);
        double r49187 = 2.0;
        double r49188 = r49187 * r49171;
        double r49189 = fma(r49184, r49186, r49188);
        double r49190 = fma(r49181, r49183, r49189);
        double r49191 = r49190 / r49176;
        double r49192 = sin(r49178);
        double r49193 = r49191 * r49192;
        double r49194 = /* ERROR: no complex support in C */;
        double r49195 = /* ERROR: no complex support in C */;
        return r49195;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))