Average Error: 43.3 → 0.8
Time: 41.2s
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 r4819346 = 0.5;
        double r4819347 = re;
        double r4819348 = sin(r4819347);
        double r4819349 = r4819346 * r4819348;
        double r4819350 = im;
        double r4819351 = -r4819350;
        double r4819352 = exp(r4819351);
        double r4819353 = exp(r4819350);
        double r4819354 = r4819352 - r4819353;
        double r4819355 = r4819349 * r4819354;
        return r4819355;
}
↓
double f(double re, double im) {
        double r4819356 = re;
        double r4819357 = sin(r4819356);
        double r4819358 = 0.5;
        double r4819359 = -r4819358;
        double r4819360 = r4819357 * r4819359;
        double r4819361 = im;
        double r4819362 = 5.0;
        double r4819363 = pow(r4819361, r4819362);
        double r4819364 = 0.016666666666666666;
        double r4819365 = 2.0;
        double r4819366 = r4819361 * r4819365;
        double r4819367 = r4819361 * r4819361;
        double r4819368 = 0.3333333333333333;
        double r4819369 = r4819367 * r4819368;
        double r4819370 = r4819361 * r4819369;
        double r4819371 = r4819366 + r4819370;
        double r4819372 = fma(r4819363, r4819364, r4819371);
        double r4819373 = r4819360 * r4819372;
        return r4819373;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.3
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.3

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

  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

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

  5. Applied distribute-lft-in0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019138 +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))))