Average Error: 0.0 → 0.0
Time: 812.0ms
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 r3344 = re;
        double r3345 = im;
        double r3346 = r3344 * r3345;
        double r3347 = r3345 * r3344;
        double r3348 = r3346 + r3347;
        return r3348;
}
↓
double f(double re, double im) {
        double r3349 = re;
        double r3350 = im;
        double r3351 = r3350 * r3349;
        double r3352 = fma(r3349, r3350, r3351);
        return r3352;
}

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 2020083 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))