Average Error: 58.1 → 0.7
Time: 38.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im - im \cdot 2\right) - \frac{1}{60} \cdot {im}^{5}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im - im \cdot 2\right) - \frac{1}{60} \cdot {im}^{5}\right)
double f(double re, double im) {
        double r6381064 = 0.5;
        double r6381065 = re;
        double r6381066 = cos(r6381065);
        double r6381067 = r6381064 * r6381066;
        double r6381068 = 0.0;
        double r6381069 = im;
        double r6381070 = r6381068 - r6381069;
        double r6381071 = exp(r6381070);
        double r6381072 = exp(r6381069);
        double r6381073 = r6381071 - r6381072;
        double r6381074 = r6381067 * r6381073;
        return r6381074;
}
↓
double f(double re, double im) {
        double r6381075 = 0.5;
        double r6381076 = re;
        double r6381077 = cos(r6381076);
        double r6381078 = r6381075 * r6381077;
        double r6381079 = im;
        double r6381080 = r6381079 * r6381079;
        double r6381081 = -0.3333333333333333;
        double r6381082 = r6381080 * r6381081;
        double r6381083 = r6381082 * r6381079;
        double r6381084 = 2.0;
        double r6381085 = r6381079 * r6381084;
        double r6381086 = r6381083 - r6381085;
        double r6381087 = 0.016666666666666666;
        double r6381088 = 5.0;
        double r6381089 = pow(r6381079, r6381088);
        double r6381090 = r6381087 * r6381089;
        double r6381091 = r6381086 - r6381090;
        double r6381092 = r6381078 * r6381091;
        return r6381092;
}

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
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))