Average Error: 0.0 → 0.0
Time: 5.8s
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 r20862 = 0.5;
        double r20863 = re;
        double r20864 = sin(r20863);
        double r20865 = r20862 * r20864;
        double r20866 = 0.0;
        double r20867 = im;
        double r20868 = r20866 - r20867;
        double r20869 = exp(r20868);
        double r20870 = exp(r20867);
        double r20871 = r20869 + r20870;
        double r20872 = r20865 * r20871;
        return r20872;
}
↓
double f(double re, double im) {
        double r20873 = 0.5;
        double r20874 = re;
        double r20875 = sin(r20874);
        double r20876 = r20873 * r20875;
        double r20877 = 0.0;
        double r20878 = im;
        double r20879 = r20877 - r20878;
        double r20880 = exp(r20879);
        double r20881 = exp(r20878);
        double r20882 = r20876 * r20881;
        double r20883 = fma(r20876, r20880, r20882);
        return r20883;
}

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))))