Average Error: 43.3 → 0.7
Time: 1.4m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60} - im \cdot \left(2 + im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60} - im \cdot \left(2 + im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r59293802 = 0.5;
        double r59293803 = re;
        double r59293804 = sin(r59293803);
        double r59293805 = r59293802 * r59293804;
        double r59293806 = im;
        double r59293807 = -r59293806;
        double r59293808 = exp(r59293807);
        double r59293809 = exp(r59293806);
        double r59293810 = r59293808 - r59293809;
        double r59293811 = r59293805 * r59293810;
        return r59293811;
}
↓
double f(double re, double im) {
        double r59293812 = im;
        double r59293813 = 5.0;
        double r59293814 = pow(r59293812, r59293813);
        double r59293815 = -0.016666666666666666;
        double r59293816 = r59293814 * r59293815;
        double r59293817 = 2.0;
        double r59293818 = 0.3333333333333333;
        double r59293819 = r59293818 * r59293812;
        double r59293820 = r59293812 * r59293819;
        double r59293821 = r59293817 + r59293820;
        double r59293822 = r59293812 * r59293821;
        double r59293823 = r59293816 - r59293822;
        double r59293824 = 0.5;
        double r59293825 = re;
        double r59293826 = sin(r59293825);
        double r59293827 = r59293824 * r59293826;
        double r59293828 = r59293823 * r59293827;
        return r59293828;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.3
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 43.3

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019119 
(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))))