Average Error: 43.4 → 0.7
Time: 39.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\sin re \cdot \left(-0.5\right)\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \left(2 + \left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)\]
\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 \left(2 + \left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)
double f(double re, double im) {
        double r9728015 = 0.5;
        double r9728016 = re;
        double r9728017 = sin(r9728016);
        double r9728018 = r9728015 * r9728017;
        double r9728019 = im;
        double r9728020 = -r9728019;
        double r9728021 = exp(r9728020);
        double r9728022 = exp(r9728019);
        double r9728023 = r9728021 - r9728022;
        double r9728024 = r9728018 * r9728023;
        return r9728024;
}
↓
double f(double re, double im) {
        double r9728025 = re;
        double r9728026 = sin(r9728025);
        double r9728027 = 0.5;
        double r9728028 = -r9728027;
        double r9728029 = r9728026 * r9728028;
        double r9728030 = im;
        double r9728031 = 5.0;
        double r9728032 = pow(r9728030, r9728031);
        double r9728033 = 0.016666666666666666;
        double r9728034 = 2.0;
        double r9728035 = r9728030 * r9728030;
        double r9728036 = 0.3333333333333333;
        double r9728037 = r9728035 * r9728036;
        double r9728038 = r9728034 + r9728037;
        double r9728039 = r9728030 * r9728038;
        double r9728040 = fma(r9728032, r9728033, r9728039);
        double r9728041 = r9728029 * r9728040;
        return r9728041;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
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 43.4

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019139 +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))))