Average Error: 43.9 → 0.8
Time: 39.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{60}, {x}^{5}, x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2063394 = x;
        double r2063395 = exp(r2063394);
        double r2063396 = -r2063394;
        double r2063397 = exp(r2063396);
        double r2063398 = r2063395 + r2063397;
        double r2063399 = 2.0;
        double r2063400 = r2063398 / r2063399;
        double r2063401 = y;
        double r2063402 = cos(r2063401);
        double r2063403 = r2063400 * r2063402;
        double r2063404 = r2063395 - r2063397;
        double r2063405 = r2063404 / r2063399;
        double r2063406 = sin(r2063401);
        double r2063407 = r2063405 * r2063406;
        double r2063408 = /* ERROR: no complex support in C */;
        double r2063409 = /* ERROR: no complex support in C */;
        return r2063409;
}
↓
double f(double x, double y) {
        double r2063410 = x;
        double r2063411 = exp(r2063410);
        double r2063412 = -r2063410;
        double r2063413 = exp(r2063412);
        double r2063414 = r2063411 + r2063413;
        double r2063415 = 2.0;
        double r2063416 = r2063414 / r2063415;
        double r2063417 = y;
        double r2063418 = cos(r2063417);
        double r2063419 = r2063416 * r2063418;
        double r2063420 = 0.016666666666666666;
        double r2063421 = 5.0;
        double r2063422 = pow(r2063410, r2063421);
        double r2063423 = 0.3333333333333333;
        double r2063424 = r2063423 * r2063410;
        double r2063425 = r2063410 * r2063424;
        double r2063426 = r2063425 + r2063415;
        double r2063427 = r2063410 * r2063426;
        double r2063428 = fma(r2063420, r2063422, r2063427);
        double r2063429 = r2063428 / r2063415;
        double r2063430 = sin(r2063417);
        double r2063431 = r2063429 * r2063430;
        double r2063432 = /* ERROR: no complex support in C */;
        double r2063433 = /* ERROR: no complex support in C */;
        return r2063433;
}

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 +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)))))