Average Error: 43.6 → 0.8
Time: 1.6m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[-\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) + im \cdot 2\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
-\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{3}\right) + im \cdot 2\right)
double f(double re, double im) {
        double r7856200 = 0.5;
        double r7856201 = re;
        double r7856202 = sin(r7856201);
        double r7856203 = r7856200 * r7856202;
        double r7856204 = im;
        double r7856205 = -r7856204;
        double r7856206 = exp(r7856205);
        double r7856207 = exp(r7856204);
        double r7856208 = r7856206 - r7856207;
        double r7856209 = r7856203 * r7856208;
        return r7856209;
}
↓
double f(double re, double im) {
        double r7856210 = 0.5;
        double r7856211 = re;
        double r7856212 = sin(r7856211);
        double r7856213 = r7856210 * r7856212;
        double r7856214 = im;
        double r7856215 = 5.0;
        double r7856216 = pow(r7856214, r7856215);
        double r7856217 = 0.016666666666666666;
        double r7856218 = r7856214 * r7856214;
        double r7856219 = 0.3333333333333333;
        double r7856220 = r7856218 * r7856219;
        double r7856221 = r7856214 * r7856220;
        double r7856222 = 2.0;
        double r7856223 = r7856214 * r7856222;
        double r7856224 = r7856221 + r7856223;
        double r7856225 = fma(r7856216, r7856217, r7856224);
        double r7856226 = r7856213 * r7856225;
        double r7856227 = -r7856226;
        return r7856227;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.6
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 43.6

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

  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

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

  5. Applied fma-udef0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))