math.square on complex, real part

Average Error: 0.0 → 0.0

Time: 1.1s

Precision: 64

Math

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

↓

\[\left(\left|im\right| + \left|re\right|\right) \cdot \left(\left|re\right| + \left(-\left|im\right|\right)\right)\]

C

double code(double re, double im) {
	return ((re * re) - (im * im));
}

↓

double code(double re, double im) {
	return ((fabs(im) + fabs(re)) * (fabs(re) + -fabs(im)));
}

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 add-sqr-sqrt 0.0

   \[\leadsto re \cdot re - \color{blue}{\sqrt{im \cdot im} \cdot \sqrt{im \cdot im}}\]

4. Applied add-sqr-sqrt 0.0

   \[\leadsto \color{blue}{\sqrt{re \cdot re} \cdot \sqrt{re \cdot re}} - \sqrt{im \cdot im} \cdot \sqrt{im \cdot im}\]

5. Applied difference-of-squares 0.0

   \[\leadsto \color{blue}{\left(\sqrt{re \cdot re} + \sqrt{im \cdot im}\right) \cdot \left(\sqrt{re \cdot re} - \sqrt{im \cdot im}\right)}\]

6. Simplified 0.0

   \[\leadsto \color{blue}{\left(\left|im\right| + \left|re\right|\right)} \cdot \left(\sqrt{re \cdot re} - \sqrt{im \cdot im}\right)\]

7. Simplified 0.0

   \[\leadsto \left(\left|im\right| + \left|re\right|\right) \cdot \color{blue}{\left(\left|re\right| + \left(-\left|im\right|\right)\right)}\]

8. Final simplification 0.0

   \[\leadsto \left(\left|im\right| + \left|re\right|\right) \cdot \left(\left|re\right| + \left(-\left|im\right|\right)\right)\]

Reproduce

herbie shell --seed 2020066 
(FPCore (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))