Average Error: 43.2 → 0.8
Time: 17.4s
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 r41535 = x;
        double r41536 = exp(r41535);
        double r41537 = -r41535;
        double r41538 = exp(r41537);
        double r41539 = r41536 + r41538;
        double r41540 = 2.0;
        double r41541 = r41539 / r41540;
        double r41542 = y;
        double r41543 = cos(r41542);
        double r41544 = r41541 * r41543;
        double r41545 = r41536 - r41538;
        double r41546 = r41545 / r41540;
        double r41547 = sin(r41542);
        double r41548 = r41546 * r41547;
        double r41549 = /* ERROR: no complex support in C */;
        double r41550 = /* ERROR: no complex support in C */;
        return r41550;
}
↓
double f(double x, double y) {
        double r41551 = 0.3333333333333333;
        double r41552 = x;
        double r41553 = 3.0;
        double r41554 = pow(r41552, r41553);
        double r41555 = r41551 * r41554;
        double r41556 = 0.016666666666666666;
        double r41557 = 5.0;
        double r41558 = pow(r41552, r41557);
        double r41559 = r41556 * r41558;
        double r41560 = 2.0;
        double r41561 = r41560 * r41552;
        double r41562 = r41559 + r41561;
        double r41563 = r41555 + r41562;
        double r41564 = 2.0;
        double r41565 = r41563 / r41564;
        double r41566 = y;
        double r41567 = sin(r41566);
        double r41568 = r41565 * r41567;
        return r41568;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

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

  3. Taylor expanded around 0 0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020046 
(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)))))