Average Error: 43.7 → 0.5
Time: 10.3s
Precision: binary64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -0.014382893694163904:\\ \;\;\;\;0.5 \cdot \frac{\sin re}{e^{im}} + e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array}\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\begin{array}{l}
\mathbf{if}\;im \le -0.014382893694163904:\\
\;\;\;\;0.5 \cdot \frac{\sin re}{e^{im}} + e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\

\end{array}
double code(double re, double im) {
	return ((double) (((double) (0.5 * ((double) sin(re)))) * ((double) (((double) exp(((double) -(im)))) - ((double) exp(im))))));
}
↓
double code(double re, double im) {
	double VAR;
	if ((im <= -0.014382893694163904)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) sin(re)) / ((double) exp(im)))))) + ((double) (((double) exp(im)) * ((double) (0.5 * ((double) -(((double) sin(re))))))))));
	} else {
		VAR = ((double) (((double) (0.5 * ((double) sin(re)))) * ((double) (((double) (im * -2.0)) + ((double) (((double) (((double) pow(im, 3.0)) * -0.3333333333333333)) + ((double) (((double) pow(im, 5.0)) * -0.016666666666666666))))))));
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < -0.014382893694163904

    1. Initial program 4.4

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
    2. Using strategy rm

    3. Applied sub-neg4.4

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{-im} + \left(-e^{im}\right)\right)}\]

    4. Applied distribute-lft-in4.6

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{-im} + \left(0.5 \cdot \sin re\right) \cdot \left(-e^{im}\right)}\]

    5. Simplified4.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin re}{e^{im}}} + \left(0.5 \cdot \sin re\right) \cdot \left(-e^{im}\right)\]

    6. Simplified4.4

      \[\leadsto 0.5 \cdot \frac{\sin re}{e^{im}} + \color{blue}{e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)}\]

    if -0.014382893694163904 < im

    1. Initial program 44.1

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(0.3333333333333333 \cdot {im}^{3} + \left(0.016666666666666666 \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

    3. Simplified0.4

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -0.014382893694163904:\\ \;\;\;\;0.5 \cdot \frac{\sin re}{e^{im}} + e^{im} \cdot \left(0.5 \cdot \left(-\sin re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))