Average Error: 43.8 → 0.7
Time: 8.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)
double f(double re, double im) {
        double r378448 = 0.5;
        double r378449 = re;
        double r378450 = sin(r378449);
        double r378451 = r378448 * r378450;
        double r378452 = im;
        double r378453 = -r378452;
        double r378454 = exp(r378453);
        double r378455 = exp(r378452);
        double r378456 = r378454 - r378455;
        double r378457 = r378451 * r378456;
        return r378457;
}
↓
double f(double re, double im) {
        double r378458 = 0.16666666666666666;
        double r378459 = re;
        double r378460 = sin(r378459);
        double r378461 = im;
        double r378462 = 3.0;
        double r378463 = pow(r378461, r378462);
        double r378464 = r378460 * r378463;
        double r378465 = r378458 * r378464;
        double r378466 = 1.0;
        double r378467 = r378460 * r378461;
        double r378468 = r378466 * r378467;
        double r378469 = 0.008333333333333333;
        double r378470 = 5.0;
        double r378471 = pow(r378461, r378470);
        double r378472 = r378460 * r378471;
        double r378473 = r378469 * r378472;
        double r378474 = r378468 + r378473;
        double r378475 = r378465 + r378474;
        double r378476 = -r378475;
        return r378476;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.8
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.8

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

  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. Taylor expanded around inf 0.7

    \[\leadsto \color{blue}{-\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto -\left(0.166666666666666657 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1 \cdot \left(\sin re \cdot im\right) + 0.00833333333333333322 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))