Average Error: 0.0 → 0.0
Time: 697.0ms
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 r2233 = re;
        double r2234 = im;
        double r2235 = r2233 * r2234;
        double r2236 = r2234 * r2233;
        double r2237 = r2235 + r2236;
        return r2237;
}
↓
double f(double re, double im) {
        double r2238 = re;
        double r2239 = im;
        double r2240 = r2239 * r2238;
        double r2241 = fma(r2238, r2239, r2240);
        return r2241;
}

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 2020033 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))