Average Error: 0.0 → 0.0
Time: 18.4s
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 r8878 = re;
        double r8879 = r8878 * r8878;
        double r8880 = im;
        double r8881 = r8880 * r8880;
        double r8882 = r8879 - r8881;
        return r8882;
}
↓
double f(double re, double im) {
        double r8883 = re;
        double r8884 = im;
        double r8885 = r8884 * r8884;
        double r8886 = -r8885;
        double r8887 = fma(r8883, r8883, r8886);
        return r8887;
}

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 2019347 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))