Average Error: 0.0 → 0.0
Time: 26.7s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\mathsf{fma}\left(re, re, -im \cdot im\right)\]
re \cdot re - im \cdot im
↓
\mathsf{fma}\left(re, re, -im \cdot im\right)
double f(double re, double im) {
        double r913345 = re;
        double r913346 = r913345 * r913345;
        double r913347 = im;
        double r913348 = r913347 * r913347;
        double r913349 = r913346 - r913348;
        return r913349;
}
↓
double f(double re, double im) {
        double r913350 = re;
        double r913351 = im;
        double r913352 = r913351 * r913351;
        double r913353 = -r913352;
        double r913354 = fma(r913350, r913350, r913353);
        return r913354;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot re - im \cdot im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))