Average Error: 43.5 → 0.7
Time: 26.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} \cdot x\right), x + x\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(x, x \cdot \left(\frac{1}{3} \cdot x\right), x + x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1033001 = x;
        double r1033002 = exp(r1033001);
        double r1033003 = -r1033001;
        double r1033004 = exp(r1033003);
        double r1033005 = r1033002 + r1033004;
        double r1033006 = 2.0;
        double r1033007 = r1033005 / r1033006;
        double r1033008 = y;
        double r1033009 = cos(r1033008);
        double r1033010 = r1033007 * r1033009;
        double r1033011 = r1033002 - r1033004;
        double r1033012 = r1033011 / r1033006;
        double r1033013 = sin(r1033008);
        double r1033014 = r1033012 * r1033013;
        double r1033015 = /* ERROR: no complex support in C */;
        double r1033016 = /* ERROR: no complex support in C */;
        return r1033016;
}
↓
double f(double x, double y) {
        double r1033017 = x;
        double r1033018 = exp(r1033017);
        double r1033019 = -r1033017;
        double r1033020 = exp(r1033019);
        double r1033021 = r1033018 + r1033020;
        double r1033022 = 2.0;
        double r1033023 = r1033021 / r1033022;
        double r1033024 = y;
        double r1033025 = cos(r1033024);
        double r1033026 = r1033023 * r1033025;
        double r1033027 = 5.0;
        double r1033028 = pow(r1033017, r1033027);
        double r1033029 = 0.016666666666666666;
        double r1033030 = 0.3333333333333333;
        double r1033031 = r1033030 * r1033017;
        double r1033032 = r1033017 * r1033031;
        double r1033033 = r1033017 + r1033017;
        double r1033034 = fma(r1033017, r1033032, r1033033);
        double r1033035 = fma(r1033028, r1033029, r1033034);
        double r1033036 = r1033035 / r1033022;
        double r1033037 = sin(r1033024);
        double r1033038 = r1033036 * r1033037;
        double r1033039 = /* ERROR: no complex support in C */;
        double r1033040 = /* ERROR: no complex support in C */;
        return r1033040;
}

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

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

  4. Taylor expanded around 0 0.7

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

  5. Simplified0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019154 +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)))))