Average Error: 0.0 → 0.0
Time: 30.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r15920 = 0.5;
        double r15921 = re;
        double r15922 = sin(r15921);
        double r15923 = r15920 * r15922;
        double r15924 = 0.0;
        double r15925 = im;
        double r15926 = r15924 - r15925;
        double r15927 = exp(r15926);
        double r15928 = exp(r15925);
        double r15929 = r15927 + r15928;
        double r15930 = r15923 * r15929;
        return r15930;
}
↓
double f(double re, double im) {
        double r15931 = im;
        double r15932 = exp(r15931);
        double r15933 = re;
        double r15934 = sin(r15933);
        double r15935 = r15934 / r15932;
        double r15936 = fma(r15932, r15934, r15935);
        double r15937 = 0.5;
        double r15938 = r15936 * r15937;
        return r15938;
}

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. Taylor expanded around inf 0.0

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

  5. Simplified0.0

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

  6. Taylor expanded around inf 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))))