math.sin on complex, real part

Average Error: 0.1 → 0.1

Time: 4.1s

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 r18098 = 0.5;
        double r18099 = re;
        double r18100 = sin(r18099);
        double r18101 = r18098 * r18100;
        double r18102 = 0.0;
        double r18103 = im;
        double r18104 = r18102 - r18103;
        double r18105 = exp(r18104);
        double r18106 = exp(r18103);
        double r18107 = r18105 + r18106;
        double r18108 = r18101 * r18107;
        return r18108;
}

↓

double f(double re, double im) {
        double r18109 = 0.5;
        double r18110 = re;
        double r18111 = sin(r18110);
        double r18112 = r18109 * r18111;
        double r18113 = 0.0;
        double r18114 = im;
        double r18115 = r18113 - r18114;
        double r18116 = exp(r18115);
        double r18117 = r18112 * r18116;
        double r18118 = exp(r18114);
        double r18119 = r18112 * r18118;
        double r18120 = r18117 + r18119;
        return r18120;
}

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