math.sin on complex, real part

Average Error: 0.0 → 0.1

Time: 5.5s

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 r71174 = 0.5;
        double r71175 = re;
        double r71176 = sin(r71175);
        double r71177 = r71174 * r71176;
        double r71178 = 0.0;
        double r71179 = im;
        double r71180 = r71178 - r71179;
        double r71181 = exp(r71180);
        double r71182 = exp(r71179);
        double r71183 = r71181 + r71182;
        double r71184 = r71177 * r71183;
        return r71184;
}

↓

double f(double re, double im) {
        double r71185 = 0.5;
        double r71186 = re;
        double r71187 = sin(r71186);
        double r71188 = r71185 * r71187;
        double r71189 = 0.0;
        double r71190 = im;
        double r71191 = r71189 - r71190;
        double r71192 = exp(r71191);
        double r71193 = r71188 * r71192;
        double r71194 = exp(r71190);
        double r71195 = r71188 * r71194;
        double r71196 = r71193 + r71195;
        return r71196;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.0

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
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 2019354 +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))))