Average Error: 43.7 → 0.8
Time: 30.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, 2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, 2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2531580 = x;
        double r2531581 = exp(r2531580);
        double r2531582 = -r2531580;
        double r2531583 = exp(r2531582);
        double r2531584 = r2531581 + r2531583;
        double r2531585 = 2.0;
        double r2531586 = r2531584 / r2531585;
        double r2531587 = y;
        double r2531588 = cos(r2531587);
        double r2531589 = r2531586 * r2531588;
        double r2531590 = r2531581 - r2531583;
        double r2531591 = r2531590 / r2531585;
        double r2531592 = sin(r2531587);
        double r2531593 = r2531591 * r2531592;
        double r2531594 = /* ERROR: no complex support in C */;
        double r2531595 = /* ERROR: no complex support in C */;
        return r2531595;
}
↓
double f(double x, double y) {
        double r2531596 = x;
        double r2531597 = exp(r2531596);
        double r2531598 = -r2531596;
        double r2531599 = exp(r2531598);
        double r2531600 = r2531597 + r2531599;
        double r2531601 = 2.0;
        double r2531602 = r2531600 / r2531601;
        double r2531603 = y;
        double r2531604 = cos(r2531603);
        double r2531605 = r2531602 * r2531604;
        double r2531606 = 5.0;
        double r2531607 = pow(r2531596, r2531606);
        double r2531608 = 0.016666666666666666;
        double r2531609 = r2531601 * r2531596;
        double r2531610 = 0.3333333333333333;
        double r2531611 = r2531596 * r2531596;
        double r2531612 = r2531610 * r2531611;
        double r2531613 = r2531612 * r2531596;
        double r2531614 = r2531609 + r2531613;
        double r2531615 = fma(r2531607, r2531608, r2531614);
        double r2531616 = r2531615 / r2531601;
        double r2531617 = sin(r2531603);
        double r2531618 = r2531616 * r2531617;
        double r2531619 = /* ERROR: no complex support in C */;
        double r2531620 = /* ERROR: no complex support in C */;
        return r2531620;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  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

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

  5. Applied distribute-rgt-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 +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)))))