Average Error: 43.7 → 0.8
Time: 1.0m
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r2310398 = x;
        double r2310399 = exp(r2310398);
        double r2310400 = -r2310398;
        double r2310401 = exp(r2310400);
        double r2310402 = r2310399 + r2310401;
        double r2310403 = 2.0;
        double r2310404 = r2310402 / r2310403;
        double r2310405 = y;
        double r2310406 = cos(r2310405);
        double r2310407 = r2310404 * r2310406;
        double r2310408 = r2310399 - r2310401;
        double r2310409 = r2310408 / r2310403;
        double r2310410 = sin(r2310405);
        double r2310411 = r2310409 * r2310410;
        double r2310412 = /* ERROR: no complex support in C */;
        double r2310413 = /* ERROR: no complex support in C */;
        return r2310413;
}
↓
double f(double x, double y) {
        double r2310414 = x;
        double r2310415 = exp(r2310414);
        double r2310416 = -r2310414;
        double r2310417 = exp(r2310416);
        double r2310418 = r2310415 + r2310417;
        double r2310419 = 2.0;
        double r2310420 = r2310418 / r2310419;
        double r2310421 = y;
        double r2310422 = cos(r2310421);
        double r2310423 = r2310420 * r2310422;
        double r2310424 = 0.016666666666666666;
        double r2310425 = 5.0;
        double r2310426 = pow(r2310414, r2310425);
        double r2310427 = r2310419 * r2310414;
        double r2310428 = r2310414 * r2310414;
        double r2310429 = 0.3333333333333333;
        double r2310430 = r2310428 * r2310429;
        double r2310431 = r2310430 * r2310414;
        double r2310432 = r2310427 + r2310431;
        double r2310433 = fma(r2310424, r2310426, r2310432);
        double r2310434 = r2310433 / r2310419;
        double r2310435 = sin(r2310421);
        double r2310436 = r2310434 * r2310435;
        double r2310437 = /* ERROR: no complex support in C */;
        double r2310438 = /* ERROR: no complex support in C */;
        return r2310438;
}

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

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

  5. Applied fma-udef0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019120 +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)))))