Average Error: 30.9 → 0.1
Time: 2.8s
Precision: binary64
\[\sqrt{2 \cdot {x}^{2}} \]
↓
\[\mathsf{hypot}\left(x, x\right) \]
\sqrt{2 \cdot {x}^{2}}
↓
\mathsf{hypot}\left(x, x\right)

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))
↓
(FPCore (x) :precision binary64 (hypot x x))
double code(double x) {
	return sqrt(2.0 * pow(x, 2.0));
}
↓
double code(double x) {
	return hypot(x, x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\sqrt{2 \cdot {x}^{2}} \]

  2. Applied add-sqr-sqrt_binary6431.1

    \[\leadsto \color{blue}{\sqrt{\sqrt{2 \cdot {x}^{2}}} \cdot \sqrt{\sqrt{2 \cdot {x}^{2}}}} \]

  3. Simplified31.1

    \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(x, x\right)}} \cdot \sqrt{\sqrt{2 \cdot {x}^{2}}} \]

  4. Simplified0.5

    \[\leadsto \sqrt{\mathsf{hypot}\left(x, x\right)} \cdot \color{blue}{\sqrt{\mathsf{hypot}\left(x, x\right)}} \]

  5. Applied pow1_binary640.5

    \[\leadsto \sqrt{\mathsf{hypot}\left(x, x\right)} \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x, x\right)}\right)}^{1}} \]

  6. Applied pow1_binary640.5

    \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x, x\right)}\right)}^{1}} \cdot {\left(\sqrt{\mathsf{hypot}\left(x, x\right)}\right)}^{1} \]

  7. Applied pow-prod-down_binary640.5

    \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt{\mathsf{hypot}\left(x, x\right)}\right)}^{1}} \]

  8. Simplified0.1

    \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x, x\right)\right)}}^{1} \]

  9. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(x, x\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))