Average Error: 57.8 → 0.8
Time: 1.3m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right))_* \cdot \left(0.5 \cdot \cos re\right)\]
double f(double re, double im) {
        double r25187076 = 0.5;
        double r25187077 = re;
        double r25187078 = cos(r25187077);
        double r25187079 = r25187076 * r25187078;
        double r25187080 = 0.0;
        double r25187081 = im;
        double r25187082 = r25187080 - r25187081;
        double r25187083 = exp(r25187082);
        double r25187084 = exp(r25187081);
        double r25187085 = r25187083 - r25187084;
        double r25187086 = r25187079 * r25187085;
        return r25187086;
}
↓
double f(double re, double im) {
        double r25187087 = im;
        double r25187088 = 5.0;
        double r25187089 = pow(r25187087, r25187088);
        double r25187090 = -0.016666666666666666;
        double r25187091 = -2.0;
        double r25187092 = r25187087 * r25187091;
        double r25187093 = -0.3333333333333333;
        double r25187094 = r25187087 * r25187093;
        double r25187095 = r25187087 * r25187094;
        double r25187096 = r25187087 * r25187095;
        double r25187097 = r25187092 + r25187096;
        double r25187098 = fma(r25187089, r25187090, r25187097);
        double r25187099 = 0.5;
        double r25187100 = re;
        double r25187101 = cos(r25187100);
        double r25187102 = r25187099 * r25187101;
        double r25187103 = r25187098 * r25187102;
        return r25187103;
}


\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right))_* \cdot \left(0.5 \cdot \cos re\right)

Error

Bits error versus re

Bits error versus im

Target

Original57.8
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.8

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

  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. Simplified0.8

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left(im \cdot \left(\left(im \cdot \frac{-1}{3}\right) \cdot im - 2\right)\right))_*}\]
  4. Using strategy rm

  5. Applied sub-neg0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(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))))