Average Error: 43.7 → 0.7
Time: 30.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(\sin re \cdot \left(-0.5\right)\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im + im\right) + \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(\sin re \cdot \left(-0.5\right)\right) + \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r4350066 = 0.5;
        double r4350067 = re;
        double r4350068 = sin(r4350067);
        double r4350069 = r4350066 * r4350068;
        double r4350070 = im;
        double r4350071 = -r4350070;
        double r4350072 = exp(r4350071);
        double r4350073 = exp(r4350070);
        double r4350074 = r4350072 - r4350073;
        double r4350075 = r4350069 * r4350074;
        return r4350075;
}
↓
double f(double re, double im) {
        double r4350076 = im;
        double r4350077 = r4350076 + r4350076;
        double r4350078 = 0.016666666666666666;
        double r4350079 = 5.0;
        double r4350080 = pow(r4350076, r4350079);
        double r4350081 = r4350078 * r4350080;
        double r4350082 = r4350077 + r4350081;
        double r4350083 = re;
        double r4350084 = sin(r4350083);
        double r4350085 = 0.5;
        double r4350086 = -r4350085;
        double r4350087 = r4350084 * r4350086;
        double r4350088 = r4350082 * r4350087;
        double r4350089 = r4350076 * r4350076;
        double r4350090 = r4350076 * r4350089;
        double r4350091 = -0.3333333333333333;
        double r4350092 = r4350090 * r4350091;
        double r4350093 = r4350085 * r4350084;
        double r4350094 = r4350092 * r4350093;
        double r4350095 = r4350088 + r4350094;
        return r4350095;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
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 43.7

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

  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

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019155 
(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))))