Average Error: 43.7 → 0.8
Time: 14.4s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\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{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49107 = x;
        double r49108 = exp(r49107);
        double r49109 = -r49107;
        double r49110 = exp(r49109);
        double r49111 = r49108 + r49110;
        double r49112 = 2.0;
        double r49113 = r49111 / r49112;
        double r49114 = y;
        double r49115 = cos(r49114);
        double r49116 = r49113 * r49115;
        double r49117 = r49108 - r49110;
        double r49118 = r49117 / r49112;
        double r49119 = sin(r49114);
        double r49120 = r49118 * r49119;
        double r49121 = /* ERROR: no complex support in C */;
        double r49122 = /* ERROR: no complex support in C */;
        return r49122;
}
↓
double f(double x, double y) {
        double r49123 = x;
        double r49124 = exp(r49123);
        double r49125 = -r49123;
        double r49126 = exp(r49125);
        double r49127 = r49124 + r49126;
        double r49128 = 2.0;
        double r49129 = r49127 / r49128;
        double r49130 = y;
        double r49131 = cos(r49130);
        double r49132 = r49129 * r49131;
        double r49133 = 0.3333333333333333;
        double r49134 = 3.0;
        double r49135 = pow(r49123, r49134);
        double r49136 = 0.016666666666666666;
        double r49137 = 5.0;
        double r49138 = pow(r49123, r49137);
        double r49139 = 2.0;
        double r49140 = r49139 * r49123;
        double r49141 = fma(r49136, r49138, r49140);
        double r49142 = fma(r49133, r49135, r49141);
        double r49143 = r49142 / r49128;
        double r49144 = sin(r49130);
        double r49145 = r49143 * r49144;
        double r49146 = /* ERROR: no complex support in C */;
        double r49147 = /* ERROR: no complex support in C */;
        return r49147;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))