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

↓

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

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

↓

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

double f(double re, double im) {
        double r492007 = 0.5;
        double r492008 = re;
        double r492009 = sin(r492008);
        double r492010 = r492007 * r492009;
        double r492011 = 0.0;
        double r492012 = im;
        double r492013 = r492011 - r492012;
        double r492014 = exp(r492013);
        double r492015 = exp(r492012);
        double r492016 = r492014 + r492015;
        double r492017 = r492010 * r492016;
        return r492017;
}

↓

double f(double re, double im) {
        double r492018 = im;
        double r492019 = exp(r492018);
        double r492020 = re;
        double r492021 = sin(r492020);
        double r492022 = 0.5;
        double r492023 = r492021 * r492022;
        double r492024 = r492023 / r492019;
        double r492025 = fma(r492019, r492023, r492024);
        return r492025;
}

Derivation

Initial program 0.0
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \sin re}\]
Taylor expanded around -inf 0.0
\[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot e^{im} + 0.5 \cdot \frac{1}{e^{im}}\right)}\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(e^{im}, \sin re \cdot 0.5, \frac{\sin re \cdot 0.5}{e^{im}}\right)}\]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(e^{im}, \sin re \cdot 0.5, \frac{\sin re \cdot 0.5}{e^{im}}\right)\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))

Error

Derivation

Reproduce