Average Error: 0.0 → 0.0
Time: 13.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right)
double f(double re, double im) {
        double r613950 = 0.5;
        double r613951 = re;
        double r613952 = sin(r613951);
        double r613953 = r613950 * r613952;
        double r613954 = 0.0;
        double r613955 = im;
        double r613956 = r613954 - r613955;
        double r613957 = exp(r613956);
        double r613958 = exp(r613955);
        double r613959 = r613957 + r613958;
        double r613960 = r613953 * r613959;
        return r613960;
}
↓
double f(double re, double im) {
        double r613961 = 0.5;
        double r613962 = im;
        double r613963 = exp(r613962);
        double r613964 = re;
        double r613965 = sin(r613964);
        double r613966 = r613965 / r613963;
        double r613967 = fma(r613963, r613965, r613966);
        double r613968 = r613961 * r613967;
        return r613968;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))