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

↓

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

\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 r554571 = 0.5;
        double r554572 = re;
        double r554573 = sin(r554572);
        double r554574 = r554571 * r554573;
        double r554575 = 0.0;
        double r554576 = im;
        double r554577 = r554575 - r554576;
        double r554578 = exp(r554577);
        double r554579 = exp(r554576);
        double r554580 = r554578 + r554579;
        double r554581 = r554574 * r554580;
        return r554581;
}

↓

double f(double re, double im) {
        double r554582 = im;
        double r554583 = exp(r554582);
        double r554584 = re;
        double r554585 = sin(r554584);
        double r554586 = 0.5;
        double r554587 = r554585 * r554586;
        double r554588 = r554587 / r554583;
        double r554589 = fma(r554583, r554587, r554588);
        return r554589;
}

Derivation

Initial program 0.0
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right) \cdot \sin re}\]
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
\[\leadsto \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot \sin re, \frac{0.5 \cdot \sin re}{e^{im}}\right)}\]
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 2019165 +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