Average Error: 43.8 → 0.7
Time: 12.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r46471 = x;
        double r46472 = exp(r46471);
        double r46473 = -r46471;
        double r46474 = exp(r46473);
        double r46475 = r46472 + r46474;
        double r46476 = 2.0;
        double r46477 = r46475 / r46476;
        double r46478 = y;
        double r46479 = cos(r46478);
        double r46480 = r46477 * r46479;
        double r46481 = r46472 - r46474;
        double r46482 = r46481 / r46476;
        double r46483 = sin(r46478);
        double r46484 = r46482 * r46483;
        double r46485 = /* ERROR: no complex support in C */;
        double r46486 = /* ERROR: no complex support in C */;
        return r46486;
}
↓
double f(double x, double y) {
        double r46487 = 0.3333333333333333;
        double r46488 = x;
        double r46489 = 3.0;
        double r46490 = pow(r46488, r46489);
        double r46491 = r46487 * r46490;
        double r46492 = 0.016666666666666666;
        double r46493 = 5.0;
        double r46494 = pow(r46488, r46493);
        double r46495 = r46492 * r46494;
        double r46496 = 2.0;
        double r46497 = r46496 * r46488;
        double r46498 = r46495 + r46497;
        double r46499 = r46491 + r46498;
        double r46500 = 2.0;
        double r46501 = r46499 / r46500;
        double r46502 = y;
        double r46503 = sin(r46502);
        double r46504 = r46501 * r46503;
        return r46504;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

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

  2. Simplified43.8

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020018 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))