math.sin on complex, real part

Average Error: 0.1 → 0.1

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r37597 = 0.5;
        double r37598 = re;
        double r37599 = sin(r37598);
        double r37600 = r37597 * r37599;
        double r37601 = 0.0;
        double r37602 = im;
        double r37603 = r37601 - r37602;
        double r37604 = exp(r37603);
        double r37605 = exp(r37602);
        double r37606 = r37604 + r37605;
        double r37607 = r37600 * r37606;
        return r37607;
}

↓

double f(double re, double im) {
        double r37608 = 0.5;
        double r37609 = re;
        double r37610 = sin(r37609);
        double r37611 = r37608 * r37610;
        double r37612 = 0.0;
        double r37613 = im;
        double r37614 = r37612 - r37613;
        double r37615 = exp(r37614);
        double r37616 = r37611 * r37615;
        double r37617 = exp(r37613);
        double r37618 = r37611 * r37617;
        double r37619 = r37616 + r37618;
        return r37619;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.1

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
2. Using strategy rm

3. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0.0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}}\]

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))