Average Error: 58.0 → 0.8
Time: 11.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\]
\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}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)
double f(double re, double im) {
        double r246399 = 0.5;
        double r246400 = re;
        double r246401 = cos(r246400);
        double r246402 = r246399 * r246401;
        double r246403 = 0.0;
        double r246404 = im;
        double r246405 = r246403 - r246404;
        double r246406 = exp(r246405);
        double r246407 = exp(r246404);
        double r246408 = r246406 - r246407;
        double r246409 = r246402 * r246408;
        return r246409;
}
↓
double f(double re, double im) {
        double r246410 = 0.5;
        double r246411 = re;
        double r246412 = cos(r246411);
        double r246413 = r246410 * r246412;
        double r246414 = 0.3333333333333333;
        double r246415 = im;
        double r246416 = 3.0;
        double r246417 = pow(r246415, r246416);
        double r246418 = r246414 * r246417;
        double r246419 = -r246418;
        double r246420 = 0.016666666666666666;
        double r246421 = 5.0;
        double r246422 = pow(r246415, r246421);
        double r246423 = 2.0;
        double r246424 = r246423 * r246415;
        double r246425 = fma(r246420, r246422, r246424);
        double r246426 = r246419 - r246425;
        double r246427 = r246413 * r246426;
        return r246427;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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))))