Average Error: 43.6 → 0.9
Time: 38.2s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \left(\sqrt{\left(x \cdot x\right) \cdot \frac{1}{3} + 2} \cdot \frac{\sqrt{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)}}{\sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}}}\right) \cdot \frac{x}{\sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}} \cdot \sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}}}\right)}{2} \cdot \sin y i\right))\]
\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}, \left(\sqrt{\left(x \cdot x\right) \cdot \frac{1}{3} + 2} \cdot \frac{\sqrt{2 \cdot 2 - \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)}}{\sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}}}\right) \cdot \frac{x}{\sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}} \cdot \sqrt[3]{\sqrt{2 - \left(x \cdot x\right) \cdot \frac{1}{3}}}}\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r1547079 = x;
        double r1547080 = exp(r1547079);
        double r1547081 = -r1547079;
        double r1547082 = exp(r1547081);
        double r1547083 = r1547080 + r1547082;
        double r1547084 = 2.0;
        double r1547085 = r1547083 / r1547084;
        double r1547086 = y;
        double r1547087 = cos(r1547086);
        double r1547088 = r1547085 * r1547087;
        double r1547089 = r1547080 - r1547082;
        double r1547090 = r1547089 / r1547084;
        double r1547091 = sin(r1547086);
        double r1547092 = r1547090 * r1547091;
        double r1547093 = /* ERROR: no complex support in C */;
        double r1547094 = /* ERROR: no complex support in C */;
        return r1547094;
}
↓
double f(double x, double y) {
        double r1547095 = x;
        double r1547096 = exp(r1547095);
        double r1547097 = -r1547095;
        double r1547098 = exp(r1547097);
        double r1547099 = r1547096 + r1547098;
        double r1547100 = 2.0;
        double r1547101 = r1547099 / r1547100;
        double r1547102 = y;
        double r1547103 = cos(r1547102);
        double r1547104 = r1547101 * r1547103;
        double r1547105 = 5.0;
        double r1547106 = pow(r1547095, r1547105);
        double r1547107 = 0.016666666666666666;
        double r1547108 = r1547095 * r1547095;
        double r1547109 = 0.3333333333333333;
        double r1547110 = r1547108 * r1547109;
        double r1547111 = r1547110 + r1547100;
        double r1547112 = sqrt(r1547111);
        double r1547113 = r1547100 * r1547100;
        double r1547114 = r1547110 * r1547110;
        double r1547115 = r1547113 - r1547114;
        double r1547116 = sqrt(r1547115);
        double r1547117 = r1547100 - r1547110;
        double r1547118 = sqrt(r1547117);
        double r1547119 = cbrt(r1547118);
        double r1547120 = r1547116 / r1547119;
        double r1547121 = r1547112 * r1547120;
        double r1547122 = r1547119 * r1547119;
        double r1547123 = r1547095 / r1547122;
        double r1547124 = r1547121 * r1547123;
        double r1547125 = fma(r1547106, r1547107, r1547124);
        double r1547126 = r1547125 / r1547100;
        double r1547127 = sin(r1547102);
        double r1547128 = r1547126 * r1547127;
        double r1547129 = /* ERROR: no complex support in C */;
        double r1547130 = /* ERROR: no complex support in C */;
        return r1547130;
}

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 0.8

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

  3. Simplified0.8

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

  5. Applied add-sqr-sqrt1.5

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

  6. Applied associate-*r*1.3

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

  8. Applied flip-+1.3

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

  9. Applied sqrt-div0.9

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

  10. Applied associate-*r/0.9

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

  12. Applied add-cube-cbrt0.9

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

  13. Applied times-frac1.0

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

  14. Applied associate-*l*0.9

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

  15. Final simplification0.9

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

Reproduce

herbie shell --seed 2019134 +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)))))