Average Error: 0.0 → 0.0
Time: 597.0ms
Precision: 64
\[re \cdot im + im \cdot re\]
↓
\[\mathsf{fma}\left(re, im, im \cdot re\right)\]
re \cdot im + im \cdot re
↓
\mathsf{fma}\left(re, im, im \cdot re\right)
double f(double re, double im) {
        double r1802 = re;
        double r1803 = im;
        double r1804 = r1802 * r1803;
        double r1805 = r1803 * r1802;
        double r1806 = r1804 + r1805;
        return r1806;
}
↓
double f(double re, double im) {
        double r1807 = re;
        double r1808 = im;
        double r1809 = r1808 * r1807;
        double r1810 = fma(r1807, r1808, r1809);
        return r1810;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot im + im \cdot re\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(re, im, im \cdot re\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(re, im, im \cdot re\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))