Average Error: 43.6 → 0.7
Time: 20.8s
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 r50082 = x;
        double r50083 = exp(r50082);
        double r50084 = -r50082;
        double r50085 = exp(r50084);
        double r50086 = r50083 + r50085;
        double r50087 = 2.0;
        double r50088 = r50086 / r50087;
        double r50089 = y;
        double r50090 = cos(r50089);
        double r50091 = r50088 * r50090;
        double r50092 = r50083 - r50085;
        double r50093 = r50092 / r50087;
        double r50094 = sin(r50089);
        double r50095 = r50093 * r50094;
        double r50096 = /* ERROR: no complex support in C */;
        double r50097 = /* ERROR: no complex support in C */;
        return r50097;
}
↓
double f(double x, double y) {
        double r50098 = x;
        double r50099 = exp(r50098);
        double r50100 = -r50098;
        double r50101 = exp(r50100);
        double r50102 = r50099 + r50101;
        double r50103 = 2.0;
        double r50104 = r50102 / r50103;
        double r50105 = y;
        double r50106 = cos(r50105);
        double r50107 = r50104 * r50106;
        double r50108 = 0.3333333333333333;
        double r50109 = 3.0;
        double r50110 = pow(r50098, r50109);
        double r50111 = 0.016666666666666666;
        double r50112 = 5.0;
        double r50113 = pow(r50098, r50112);
        double r50114 = 2.0;
        double r50115 = r50114 * r50098;
        double r50116 = fma(r50111, r50113, r50115);
        double r50117 = fma(r50108, r50110, r50116);
        double r50118 = r50117 / r50103;
        double r50119 = sin(r50105);
        double r50120 = r50118 * r50119;
        double r50121 = /* ERROR: no complex support in C */;
        double r50122 = /* ERROR: no complex support in C */;
        return r50122;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020045 +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)))))