Average Error: 43.5 → 0.7
Time: 31.9s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{2 \cdot x + \left(\frac{1}{60} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{2 \cdot x + \left(\frac{1}{60} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1239635 = x;
        double r1239636 = exp(r1239635);
        double r1239637 = -r1239635;
        double r1239638 = exp(r1239637);
        double r1239639 = r1239636 + r1239638;
        double r1239640 = 2.0;
        double r1239641 = r1239639 / r1239640;
        double r1239642 = y;
        double r1239643 = cos(r1239642);
        double r1239644 = r1239641 * r1239643;
        double r1239645 = r1239636 - r1239638;
        double r1239646 = r1239645 / r1239640;
        double r1239647 = sin(r1239642);
        double r1239648 = r1239646 * r1239647;
        double r1239649 = /* ERROR: no complex support in C */;
        double r1239650 = /* ERROR: no complex support in C */;
        return r1239650;
}
↓
double f(double x, double y) {
        double r1239651 = x;
        double r1239652 = exp(r1239651);
        double r1239653 = -r1239651;
        double r1239654 = exp(r1239653);
        double r1239655 = r1239652 + r1239654;
        double r1239656 = 2.0;
        double r1239657 = r1239655 / r1239656;
        double r1239658 = y;
        double r1239659 = cos(r1239658);
        double r1239660 = r1239657 * r1239659;
        double r1239661 = r1239656 * r1239651;
        double r1239662 = 0.016666666666666666;
        double r1239663 = r1239651 * r1239651;
        double r1239664 = r1239663 * r1239651;
        double r1239665 = r1239663 * r1239664;
        double r1239666 = r1239662 * r1239665;
        double r1239667 = 0.3333333333333333;
        double r1239668 = r1239667 * r1239651;
        double r1239669 = r1239663 * r1239668;
        double r1239670 = r1239666 + r1239669;
        double r1239671 = r1239661 + r1239670;
        double r1239672 = r1239671 / r1239656;
        double r1239673 = sin(r1239658);
        double r1239674 = r1239672 * r1239673;
        double r1239675 = /* ERROR: no complex support in C */;
        double r1239676 = /* ERROR: no complex support in C */;
        return r1239676;
}

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

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

  5. Applied *-un-lft-identity0.7

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

  6. Applied associate-*r*0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019138 
(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)))))