Average Error: 43.4 → 0.8
Time: 49.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(2 + \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{3}}\right)\right)\right)}{2} \cdot \sin y i\right))\]
\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(2 + \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{3}}\right)\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1198549 = x;
        double r1198550 = exp(r1198549);
        double r1198551 = -r1198549;
        double r1198552 = exp(r1198551);
        double r1198553 = r1198550 + r1198552;
        double r1198554 = 2.0;
        double r1198555 = r1198553 / r1198554;
        double r1198556 = y;
        double r1198557 = cos(r1198556);
        double r1198558 = r1198555 * r1198557;
        double r1198559 = r1198550 - r1198552;
        double r1198560 = r1198559 / r1198554;
        double r1198561 = sin(r1198556);
        double r1198562 = r1198560 * r1198561;
        double r1198563 = /* ERROR: no complex support in C */;
        double r1198564 = /* ERROR: no complex support in C */;
        return r1198564;
}
↓
double f(double x, double y) {
        double r1198565 = x;
        double r1198566 = exp(r1198565);
        double r1198567 = -r1198565;
        double r1198568 = exp(r1198567);
        double r1198569 = r1198566 + r1198568;
        double r1198570 = 2.0;
        double r1198571 = r1198569 / r1198570;
        double r1198572 = y;
        double r1198573 = cos(r1198572);
        double r1198574 = r1198571 * r1198573;
        double r1198575 = 0.016666666666666666;
        double r1198576 = 5.0;
        double r1198577 = pow(r1198565, r1198576);
        double r1198578 = r1198565 * r1198565;
        double r1198579 = 0.3333333333333333;
        double r1198580 = r1198578 * r1198579;
        double r1198581 = exp(r1198580);
        double r1198582 = log(r1198581);
        double r1198583 = r1198570 + r1198582;
        double r1198584 = r1198565 * r1198583;
        double r1198585 = fma(r1198575, r1198577, r1198584);
        double r1198586 = r1198585 / r1198570;
        double r1198587 = sin(r1198572);
        double r1198588 = r1198586 * r1198587;
        double r1198589 = /* ERROR: no complex support in C */;
        double r1198590 = /* ERROR: no complex support in C */;
        return r1198590;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

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

  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

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

  5. Applied add-log-exp0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019142 +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)))))