Average Error: 0.0 → 0.0
Time: 25.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r1272715 = 0.5;
        double r1272716 = re;
        double r1272717 = sin(r1272716);
        double r1272718 = r1272715 * r1272717;
        double r1272719 = 0.0;
        double r1272720 = im;
        double r1272721 = r1272719 - r1272720;
        double r1272722 = exp(r1272721);
        double r1272723 = exp(r1272720);
        double r1272724 = r1272722 + r1272723;
        double r1272725 = r1272718 * r1272724;
        return r1272725;
}
↓
double f(double re, double im) {
        double r1272726 = re;
        double r1272727 = sin(r1272726);
        double r1272728 = im;
        double r1272729 = exp(r1272728);
        double r1272730 = r1272727 / r1272729;
        double r1272731 = fma(r1272727, r1272729, r1272730);
        double r1272732 = 0.5;
        double r1272733 = r1272731 * r1272732;
        return r1272733;
}

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. 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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))