Average Error: 43.6 → 0.7
Time: 31.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\sin re \cdot \left(-0.5\right)\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot 2 + im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\sin re \cdot \left(-0.5\right)\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot 2 + im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)
double f(double re, double im) {
        double r8395308 = 0.5;
        double r8395309 = re;
        double r8395310 = sin(r8395309);
        double r8395311 = r8395308 * r8395310;
        double r8395312 = im;
        double r8395313 = -r8395312;
        double r8395314 = exp(r8395313);
        double r8395315 = exp(r8395312);
        double r8395316 = r8395314 - r8395315;
        double r8395317 = r8395311 * r8395316;
        return r8395317;
}
↓
double f(double re, double im) {
        double r8395318 = re;
        double r8395319 = sin(r8395318);
        double r8395320 = 0.5;
        double r8395321 = -r8395320;
        double r8395322 = r8395319 * r8395321;
        double r8395323 = im;
        double r8395324 = 5.0;
        double r8395325 = pow(r8395323, r8395324);
        double r8395326 = 0.016666666666666666;
        double r8395327 = 2.0;
        double r8395328 = r8395323 * r8395327;
        double r8395329 = r8395323 * r8395323;
        double r8395330 = 0.3333333333333333;
        double r8395331 = r8395329 * r8395330;
        double r8395332 = r8395323 * r8395331;
        double r8395333 = r8395328 + r8395332;
        double r8395334 = fma(r8395325, r8395326, r8395333);
        double r8395335 = r8395322 * r8395334;
        return r8395335;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.6
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \mathsf{fma}\left(\frac{1}{3}, im \cdot im, 2\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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