Average Error: 58.1 → 0.8
Time: 8.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r258939 = 0.5;
        double r258940 = re;
        double r258941 = cos(r258940);
        double r258942 = r258939 * r258941;
        double r258943 = 0.0;
        double r258944 = im;
        double r258945 = r258943 - r258944;
        double r258946 = exp(r258945);
        double r258947 = exp(r258944);
        double r258948 = r258946 - r258947;
        double r258949 = r258942 * r258948;
        return r258949;
}
↓
double f(double re, double im) {
        double r258950 = 0.5;
        double r258951 = re;
        double r258952 = cos(r258951);
        double r258953 = r258950 * r258952;
        double r258954 = 0.3333333333333333;
        double r258955 = im;
        double r258956 = 3.0;
        double r258957 = pow(r258955, r258956);
        double r258958 = r258954 * r258957;
        double r258959 = 0.016666666666666666;
        double r258960 = 5.0;
        double r258961 = pow(r258955, r258960);
        double r258962 = r258959 * r258961;
        double r258963 = 2.0;
        double r258964 = r258963 * r258955;
        double r258965 = r258962 + r258964;
        double r258966 = r258958 + r258965;
        double r258967 = -r258966;
        double r258968 = r258953 * r258967;
        return r258968;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))