Average Error: 43.5 → 0.8
Time: 48.5s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, 2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r992482 = x;
        double r992483 = exp(r992482);
        double r992484 = -r992482;
        double r992485 = exp(r992484);
        double r992486 = r992483 + r992485;
        double r992487 = 2.0;
        double r992488 = r992486 / r992487;
        double r992489 = y;
        double r992490 = cos(r992489);
        double r992491 = r992488 * r992490;
        double r992492 = r992483 - r992485;
        double r992493 = r992492 / r992487;
        double r992494 = sin(r992489);
        double r992495 = r992493 * r992494;
        double r992496 = /* ERROR: no complex support in C */;
        double r992497 = /* ERROR: no complex support in C */;
        return r992497;
}
↓
double f(double x, double y) {
        double r992498 = x;
        double r992499 = exp(r992498);
        double r992500 = -r992498;
        double r992501 = exp(r992500);
        double r992502 = r992499 + r992501;
        double r992503 = 2.0;
        double r992504 = r992502 / r992503;
        double r992505 = y;
        double r992506 = cos(r992505);
        double r992507 = r992504 * r992506;
        double r992508 = 5.0;
        double r992509 = pow(r992498, r992508);
        double r992510 = 0.016666666666666666;
        double r992511 = r992503 * r992498;
        double r992512 = r992498 * r992498;
        double r992513 = 0.3333333333333333;
        double r992514 = r992512 * r992513;
        double r992515 = r992514 * r992498;
        double r992516 = r992511 + r992515;
        double r992517 = fma(r992509, r992510, r992516);
        double r992518 = r992517 / r992503;
        double r992519 = sin(r992505);
        double r992520 = r992518 * r992519;
        double r992521 = /* ERROR: no complex support in C */;
        double r992522 = /* ERROR: no complex support in C */;
        return r992522;
}

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 0.8

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

  3. Simplified0.8

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

  5. Applied distribute-lft-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019144 +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)))))