Average Error: 44.0 → 0.8
Time: 12.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\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
double f(double x, double y) {
        double r39258 = x;
        double r39259 = exp(r39258);
        double r39260 = -r39258;
        double r39261 = exp(r39260);
        double r39262 = r39259 + r39261;
        double r39263 = 2.0;
        double r39264 = r39262 / r39263;
        double r39265 = y;
        double r39266 = cos(r39265);
        double r39267 = r39264 * r39266;
        double r39268 = r39259 - r39261;
        double r39269 = r39268 / r39263;
        double r39270 = sin(r39265);
        double r39271 = r39269 * r39270;
        double r39272 = /* ERROR: no complex support in C */;
        double r39273 = /* ERROR: no complex support in C */;
        return r39273;
}
↓
double f(double x, double y) {
        double r39274 = 0.3333333333333333;
        double r39275 = x;
        double r39276 = 3.0;
        double r39277 = pow(r39275, r39276);
        double r39278 = 0.016666666666666666;
        double r39279 = 5.0;
        double r39280 = pow(r39275, r39279);
        double r39281 = 2.0;
        double r39282 = r39281 * r39275;
        double r39283 = fma(r39278, r39280, r39282);
        double r39284 = fma(r39274, r39277, r39283);
        double r39285 = 2.0;
        double r39286 = r39284 / r39285;
        double r39287 = y;
        double r39288 = sin(r39287);
        double r39289 = r39286 * r39288;
        return r39289;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 44.0

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

  2. Simplified44.0

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2020027 +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)))))