Average Error: 43.7 → 0.8
Time: 23.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} - {im}^{5} \cdot \frac{1}{60}\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(im \cdot -2 + \left({im}^{3} \cdot \frac{-1}{3} - {im}^{5} \cdot \frac{1}{60}\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r129151 = 0.5;
        double r129152 = re;
        double r129153 = sin(r129152);
        double r129154 = r129151 * r129153;
        double r129155 = im;
        double r129156 = -r129155;
        double r129157 = exp(r129156);
        double r129158 = exp(r129155);
        double r129159 = r129157 - r129158;
        double r129160 = r129154 * r129159;
        return r129160;
}
↓
double f(double re, double im) {
        double r129161 = im;
        double r129162 = -2.0;
        double r129163 = r129161 * r129162;
        double r129164 = 3.0;
        double r129165 = pow(r129161, r129164);
        double r129166 = -0.3333333333333333;
        double r129167 = r129165 * r129166;
        double r129168 = 5.0;
        double r129169 = pow(r129161, r129168);
        double r129170 = 0.016666666666666666;
        double r129171 = r129169 * r129170;
        double r129172 = r129167 - r129171;
        double r129173 = r129163 + r129172;
        double r129174 = 0.5;
        double r129175 = re;
        double r129176 = sin(r129175);
        double r129177 = r129174 * r129176;
        double r129178 = r129173 * r129177;
        return r129178;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + \left(im + im\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied pow10.8

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

  6. Applied pow10.8

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

  7. Applied pow10.8

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

  8. Applied pow-prod-down0.8

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

  9. Applied pow-prod-down0.8

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

  10. Simplified0.8

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

  11. Final simplification0.8

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

Reproduce

herbie shell --seed 2019194 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :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))))