Average Error: 43.4 → 0.7
Time: 42.3s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) + 2 \cdot x\right)}{2} \cdot \sin y i\right))\]
\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}, x \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) + 2 \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1674738 = x;
        double r1674739 = exp(r1674738);
        double r1674740 = -r1674738;
        double r1674741 = exp(r1674740);
        double r1674742 = r1674739 + r1674741;
        double r1674743 = 2.0;
        double r1674744 = r1674742 / r1674743;
        double r1674745 = y;
        double r1674746 = cos(r1674745);
        double r1674747 = r1674744 * r1674746;
        double r1674748 = r1674739 - r1674741;
        double r1674749 = r1674748 / r1674743;
        double r1674750 = sin(r1674745);
        double r1674751 = r1674749 * r1674750;
        double r1674752 = /* ERROR: no complex support in C */;
        double r1674753 = /* ERROR: no complex support in C */;
        return r1674753;
}
↓
double f(double x, double y) {
        double r1674754 = x;
        double r1674755 = exp(r1674754);
        double r1674756 = -r1674754;
        double r1674757 = exp(r1674756);
        double r1674758 = r1674755 + r1674757;
        double r1674759 = 2.0;
        double r1674760 = r1674758 / r1674759;
        double r1674761 = y;
        double r1674762 = cos(r1674761);
        double r1674763 = r1674760 * r1674762;
        double r1674764 = 0.016666666666666666;
        double r1674765 = 5.0;
        double r1674766 = pow(r1674754, r1674765);
        double r1674767 = 0.3333333333333333;
        double r1674768 = r1674754 * r1674767;
        double r1674769 = r1674768 * r1674754;
        double r1674770 = r1674754 * r1674769;
        double r1674771 = r1674759 * r1674754;
        double r1674772 = r1674770 + r1674771;
        double r1674773 = fma(r1674764, r1674766, r1674772);
        double r1674774 = r1674773 / r1674759;
        double r1674775 = sin(r1674761);
        double r1674776 = r1674774 * r1674775;
        double r1674777 = /* ERROR: no complex support in C */;
        double r1674778 = /* ERROR: no complex support in C */;
        return r1674778;
}

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 0.7

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

  3. Simplified0.7

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

  5. Applied distribute-rgt-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019139 +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)))))