Average Error: 43.9 → 0.8
Time: 54.0s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} i\right))\]
\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 + \sin y \cdot \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2} i\right))
double f(double x, double y) {
        double r2172206 = x;
        double r2172207 = exp(r2172206);
        double r2172208 = -r2172206;
        double r2172209 = exp(r2172208);
        double r2172210 = r2172207 + r2172209;
        double r2172211 = 2.0;
        double r2172212 = r2172210 / r2172211;
        double r2172213 = y;
        double r2172214 = cos(r2172213);
        double r2172215 = r2172212 * r2172214;
        double r2172216 = r2172207 - r2172209;
        double r2172217 = r2172216 / r2172211;
        double r2172218 = sin(r2172213);
        double r2172219 = r2172217 * r2172218;
        double r2172220 = /* ERROR: no complex support in C */;
        double r2172221 = /* ERROR: no complex support in C */;
        return r2172221;
}
↓
double f(double x, double y) {
        double r2172222 = x;
        double r2172223 = exp(r2172222);
        double r2172224 = -r2172222;
        double r2172225 = exp(r2172224);
        double r2172226 = r2172223 + r2172225;
        double r2172227 = 2.0;
        double r2172228 = r2172226 / r2172227;
        double r2172229 = y;
        double r2172230 = cos(r2172229);
        double r2172231 = r2172228 * r2172230;
        double r2172232 = sin(r2172229);
        double r2172233 = 0.016666666666666666;
        double r2172234 = 5.0;
        double r2172235 = pow(r2172222, r2172234);
        double r2172236 = r2172233 * r2172235;
        double r2172237 = 2.0;
        double r2172238 = r2172222 * r2172237;
        double r2172239 = 0.3333333333333333;
        double r2172240 = r2172222 * r2172222;
        double r2172241 = r2172239 * r2172240;
        double r2172242 = r2172222 * r2172241;
        double r2172243 = r2172238 + r2172242;
        double r2172244 = r2172236 + r2172243;
        double r2172245 = r2172244 / r2172227;
        double r2172246 = r2172232 * r2172245;
        double r2172247 = /* ERROR: no complex support in C */;
        double r2172248 = /* ERROR: no complex support in C */;
        return r2172248;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.9

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

  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

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

  5. Applied distribute-lft-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))