Average Error: 0.0 → 0.0
Time: 5.4s
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 r8156 = re;
        double r8157 = r8156 * r8156;
        double r8158 = im;
        double r8159 = r8158 * r8158;
        double r8160 = r8157 - r8159;
        return r8160;
}
↓
double f(double re, double im) {
        double r8161 = re;
        double r8162 = im;
        double r8163 = r8162 * r8162;
        double r8164 = -r8163;
        double r8165 = fma(r8161, r8161, r8164);
        return r8165;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot re - im \cdot im\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))