Average Error: 43.5 → 0.8
Time: 38.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), x \cdot x\right) + 2}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) \cdot x\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), x \cdot x\right) + 2}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) \cdot x\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1680527 = x;
        double r1680528 = exp(r1680527);
        double r1680529 = -r1680527;
        double r1680530 = exp(r1680529);
        double r1680531 = r1680528 + r1680530;
        double r1680532 = 2.0;
        double r1680533 = r1680531 / r1680532;
        double r1680534 = y;
        double r1680535 = cos(r1680534);
        double r1680536 = r1680533 * r1680535;
        double r1680537 = r1680528 - r1680530;
        double r1680538 = r1680537 / r1680532;
        double r1680539 = sin(r1680534);
        double r1680540 = r1680538 * r1680539;
        double r1680541 = /* ERROR: no complex support in C */;
        double r1680542 = /* ERROR: no complex support in C */;
        return r1680542;
}
↓
double f(double x, double y) {
        double r1680543 = 0.08333333333333333;
        double r1680544 = x;
        double r1680545 = r1680544 * r1680544;
        double r1680546 = r1680545 * r1680545;
        double r1680547 = fma(r1680543, r1680546, r1680545);
        double r1680548 = 2.0;
        double r1680549 = r1680547 + r1680548;
        double r1680550 = 2.0;
        double r1680551 = r1680549 / r1680550;
        double r1680552 = y;
        double r1680553 = cos(r1680552);
        double r1680554 = r1680551 * r1680553;
        double r1680555 = 0.016666666666666666;
        double r1680556 = 5.0;
        double r1680557 = pow(r1680544, r1680556);
        double r1680558 = 0.3333333333333333;
        double r1680559 = r1680558 * r1680545;
        double r1680560 = r1680559 + r1680548;
        double r1680561 = r1680560 * r1680544;
        double r1680562 = fma(r1680555, r1680557, r1680561);
        double r1680563 = r1680562 / r1680550;
        double r1680564 = sin(r1680552);
        double r1680565 = r1680563 * r1680564;
        double r1680566 = /* ERROR: no complex support in C */;
        double r1680567 = /* ERROR: no complex support in C */;
        return r1680567;
}

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 0.8

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

  3. Simplified0.8

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

  4. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{\color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))\]

  5. Simplified0.8

    \[\leadsto \Im(\left(\frac{\color{blue}{2 + \mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), x \cdot x\right)}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3} + 2\right)\right)}{2} \cdot \sin y i\right))\]

  6. Final simplification0.8

    \[\leadsto \Im(\left(\frac{\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), x \cdot x\right) + 2}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) \cdot x\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))