Average Error: 0.0 → 0.0
Time: 23.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)
double f(double re, double im) {
        double r17273 = 0.5;
        double r17274 = re;
        double r17275 = sin(r17274);
        double r17276 = r17273 * r17275;
        double r17277 = 0.0;
        double r17278 = im;
        double r17279 = r17277 - r17278;
        double r17280 = exp(r17279);
        double r17281 = exp(r17278);
        double r17282 = r17280 + r17281;
        double r17283 = r17276 * r17282;
        return r17283;
}
↓
double f(double re, double im) {
        double r17284 = 0.5;
        double r17285 = re;
        double r17286 = sin(r17285);
        double r17287 = r17284 * r17286;
        double r17288 = 0.0;
        double r17289 = im;
        double r17290 = r17288 - r17289;
        double r17291 = exp(r17290);
        double r17292 = exp(r17289);
        double r17293 = r17287 * r17292;
        double r17294 = fma(r17287, r17291, r17293);
        return r17294;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0.0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}}\]
  4. Using strategy rm

  5. Applied fma-def0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)}\]

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))