Average Error: 43.5 → 0.7
Time: 33.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(x \cdot \left(\frac{1}{3} \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(x \cdot \left(\frac{1}{3} \cdot x\right)\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1971565 = x;
        double r1971566 = exp(r1971565);
        double r1971567 = -r1971565;
        double r1971568 = exp(r1971567);
        double r1971569 = r1971566 + r1971568;
        double r1971570 = 2.0;
        double r1971571 = r1971569 / r1971570;
        double r1971572 = y;
        double r1971573 = cos(r1971572);
        double r1971574 = r1971571 * r1971573;
        double r1971575 = r1971566 - r1971568;
        double r1971576 = r1971575 / r1971570;
        double r1971577 = sin(r1971572);
        double r1971578 = r1971576 * r1971577;
        double r1971579 = /* ERROR: no complex support in C */;
        double r1971580 = /* ERROR: no complex support in C */;
        return r1971580;
}
↓
double f(double x, double y) {
        double r1971581 = x;
        double r1971582 = exp(r1971581);
        double r1971583 = -r1971581;
        double r1971584 = exp(r1971583);
        double r1971585 = r1971582 + r1971584;
        double r1971586 = 2.0;
        double r1971587 = r1971585 / r1971586;
        double r1971588 = y;
        double r1971589 = cos(r1971588);
        double r1971590 = r1971587 * r1971589;
        double r1971591 = 0.016666666666666666;
        double r1971592 = 5.0;
        double r1971593 = pow(r1971581, r1971592);
        double r1971594 = r1971586 * r1971581;
        double r1971595 = 0.3333333333333333;
        double r1971596 = r1971595 * r1971581;
        double r1971597 = r1971581 * r1971596;
        double r1971598 = r1971597 * r1971581;
        double r1971599 = r1971594 + r1971598;
        double r1971600 = fma(r1971591, r1971593, r1971599);
        double r1971601 = r1971600 / r1971586;
        double r1971602 = sin(r1971588);
        double r1971603 = r1971601 * r1971602;
        double r1971604 = /* ERROR: no complex support in C */;
        double r1971605 = /* ERROR: no complex support in C */;
        return r1971605;
}

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 0.7

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

  3. Simplified0.7

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

  5. Applied distribute-rgt-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 +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)))))