Average Error: 44.6 → 0.7
Time: 38.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(2 \cdot im + \frac{1}{60} \cdot {im}^{5}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(2 \cdot im + \frac{1}{60} \cdot {im}^{5}\right)\right)
double f(double re, double im) {
        double r12311192 = 0.5;
        double r12311193 = re;
        double r12311194 = sin(r12311193);
        double r12311195 = r12311192 * r12311194;
        double r12311196 = im;
        double r12311197 = -r12311196;
        double r12311198 = exp(r12311197);
        double r12311199 = exp(r12311196);
        double r12311200 = r12311198 - r12311199;
        double r12311201 = r12311195 * r12311200;
        return r12311201;
}
↓
double f(double re, double im) {
        double r12311202 = 0.5;
        double r12311203 = re;
        double r12311204 = sin(r12311203);
        double r12311205 = r12311202 * r12311204;
        double r12311206 = im;
        double r12311207 = r12311206 * r12311206;
        double r12311208 = r12311206 * r12311207;
        double r12311209 = -0.3333333333333333;
        double r12311210 = r12311208 * r12311209;
        double r12311211 = 2.0;
        double r12311212 = r12311211 * r12311206;
        double r12311213 = 0.016666666666666666;
        double r12311214 = 5.0;
        double r12311215 = pow(r12311206, r12311214);
        double r12311216 = r12311213 * r12311215;
        double r12311217 = r12311212 + r12311216;
        double r12311218 = r12311210 - r12311217;
        double r12311219 = r12311205 * r12311218;
        return r12311219;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original44.6
Target0.2
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 44.6

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019129 
(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))))