Average Error: 43.2 → 0.8
Time: 1.5m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r44135079 = 0.5;
        double r44135080 = re;
        double r44135081 = sin(r44135080);
        double r44135082 = r44135079 * r44135081;
        double r44135083 = im;
        double r44135084 = -r44135083;
        double r44135085 = exp(r44135084);
        double r44135086 = exp(r44135083);
        double r44135087 = r44135085 - r44135086;
        double r44135088 = r44135082 * r44135087;
        return r44135088;
}
↓
double f(double re, double im) {
        double r44135089 = im;
        double r44135090 = 5.0;
        double r44135091 = pow(r44135089, r44135090);
        double r44135092 = -0.016666666666666666;
        double r44135093 = -2.0;
        double r44135094 = r44135089 * r44135093;
        double r44135095 = -0.3333333333333333;
        double r44135096 = r44135089 * r44135095;
        double r44135097 = r44135089 * r44135096;
        double r44135098 = r44135089 * r44135097;
        double r44135099 = r44135094 + r44135098;
        double r44135100 = fma(r44135091, r44135092, r44135099);
        double r44135101 = 0.5;
        double r44135102 = re;
        double r44135103 = sin(r44135102);
        double r44135104 = r44135101 * r44135103;
        double r44135105 = r44135100 * r44135104;
        return r44135105;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.2
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(\left(im \cdot \frac{-1}{3}\right) \cdot im - 2\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied sub-neg0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(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))))