Average Error: 58.1 → 0.7
Time: 10.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r154207 = 0.5;
        double r154208 = re;
        double r154209 = cos(r154208);
        double r154210 = r154207 * r154209;
        double r154211 = 0.0;
        double r154212 = im;
        double r154213 = r154211 - r154212;
        double r154214 = exp(r154213);
        double r154215 = exp(r154212);
        double r154216 = r154214 - r154215;
        double r154217 = r154210 * r154216;
        return r154217;
}
↓
double f(double re, double im) {
        double r154218 = 0.5;
        double r154219 = re;
        double r154220 = cos(r154219);
        double r154221 = r154218 * r154220;
        double r154222 = 0.3333333333333333;
        double r154223 = im;
        double r154224 = 3.0;
        double r154225 = pow(r154223, r154224);
        double r154226 = r154222 * r154225;
        double r154227 = 0.016666666666666666;
        double r154228 = 5.0;
        double r154229 = pow(r154223, r154228);
        double r154230 = r154227 * r154229;
        double r154231 = 2.0;
        double r154232 = r154231 * r154223;
        double r154233 = r154230 + r154232;
        double r154234 = r154226 + r154233;
        double r154235 = -r154234;
        double r154236 = r154221 * r154235;
        return r154236;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Final simplification0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))