Average Error: 0.0 → 0.0
Time: 28.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r604165 = 0.5;
        double r604166 = re;
        double r604167 = sin(r604166);
        double r604168 = r604165 * r604167;
        double r604169 = 0.0;
        double r604170 = im;
        double r604171 = r604169 - r604170;
        double r604172 = exp(r604171);
        double r604173 = exp(r604170);
        double r604174 = r604172 + r604173;
        double r604175 = r604168 * r604174;
        return r604175;
}
↓
double f(double re, double im) {
        double r604176 = re;
        double r604177 = sin(r604176);
        double r604178 = 0.5;
        double r604179 = im;
        double r604180 = exp(r604179);
        double r604181 = r604178 / r604180;
        double r604182 = fma(r604178, r604180, r604181);
        double r604183 = r604177 * r604182;
        return r604183;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \sin re}\]

  3. Final simplification0.0

    \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))