math.sin on complex, real part

Average Error: 0.1 → 0.1

Time: 4.4s

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 r15434 = 0.5;
        double r15435 = re;
        double r15436 = sin(r15435);
        double r15437 = r15434 * r15436;
        double r15438 = 0.0;
        double r15439 = im;
        double r15440 = r15438 - r15439;
        double r15441 = exp(r15440);
        double r15442 = exp(r15439);
        double r15443 = r15441 + r15442;
        double r15444 = r15437 * r15443;
        return r15444;
}

↓

double f(double re, double im) {
        double r15445 = 0.5;
        double r15446 = re;
        double r15447 = sin(r15446);
        double r15448 = r15445 * r15447;
        double r15449 = 0.0;
        double r15450 = im;
        double r15451 = r15449 - r15450;
        double r15452 = exp(r15451);
        double r15453 = r15448 * r15452;
        double r15454 = exp(r15450);
        double r15455 = r15448 * r15454;
        double r15456 = r15453 + r15455;
        return r15456;
}

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 2019362 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))