Average Error: 0.0 → 0.0
Time: 16.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r206048 = 0.5;
        double r206049 = re;
        double r206050 = sin(r206049);
        double r206051 = r206048 * r206050;
        double r206052 = 0.0;
        double r206053 = im;
        double r206054 = r206052 - r206053;
        double r206055 = exp(r206054);
        double r206056 = exp(r206053);
        double r206057 = r206055 + r206056;
        double r206058 = r206051 * r206057;
        return r206058;
}
↓
double f(double re, double im) {
        double r206059 = re;
        double r206060 = sin(r206059);
        double r206061 = im;
        double r206062 = exp(r206061);
        double r206063 = 0.5;
        double r206064 = r206063 / r206062;
        double r206065 = fma(r206062, r206063, r206064);
        double r206066 = r206060 * r206065;
        return r206066;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))