Average Error: 43.5 → 0.7
Time: 34.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\frac{1}{60} \cdot {x}^{5} + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\frac{1}{60} \cdot {x}^{5} + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2340584 = x;
        double r2340585 = exp(r2340584);
        double r2340586 = -r2340584;
        double r2340587 = exp(r2340586);
        double r2340588 = r2340585 + r2340587;
        double r2340589 = 2.0;
        double r2340590 = r2340588 / r2340589;
        double r2340591 = y;
        double r2340592 = cos(r2340591);
        double r2340593 = r2340590 * r2340592;
        double r2340594 = r2340585 - r2340587;
        double r2340595 = r2340594 / r2340589;
        double r2340596 = sin(r2340591);
        double r2340597 = r2340595 * r2340596;
        double r2340598 = /* ERROR: no complex support in C */;
        double r2340599 = /* ERROR: no complex support in C */;
        return r2340599;
}
↓
double f(double x, double y) {
        double r2340600 = x;
        double r2340601 = exp(r2340600);
        double r2340602 = -r2340600;
        double r2340603 = exp(r2340602);
        double r2340604 = r2340601 + r2340603;
        double r2340605 = 2.0;
        double r2340606 = r2340604 / r2340605;
        double r2340607 = y;
        double r2340608 = cos(r2340607);
        double r2340609 = r2340606 * r2340608;
        double r2340610 = 0.016666666666666666;
        double r2340611 = 5.0;
        double r2340612 = pow(r2340600, r2340611);
        double r2340613 = r2340610 * r2340612;
        double r2340614 = 0.3333333333333333;
        double r2340615 = r2340600 * r2340600;
        double r2340616 = r2340614 * r2340615;
        double r2340617 = r2340616 * r2340600;
        double r2340618 = r2340605 * r2340600;
        double r2340619 = r2340617 + r2340618;
        double r2340620 = r2340613 + r2340619;
        double r2340621 = r2340620 / r2340605;
        double r2340622 = sin(r2340607);
        double r2340623 = r2340621 * r2340622;
        double r2340624 = /* ERROR: no complex support in C */;
        double r2340625 = /* ERROR: no complex support in C */;
        return r2340625;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

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

  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

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

  5. Applied distribute-rgt-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 
(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)))))