Average Error: 30.3 → 0.1
Time: 4.0s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)} \]
↓
\[\mathsf{hypot}\left(x, x\right) \]
\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\mathsf{hypot}\left(x, x\right)

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

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.3

    \[\sqrt{2 \cdot \left(x \cdot x\right)} \]

  2. Applied add-sqr-sqrt_binary6430.5

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

  3. Simplified30.5

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

  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 2021344 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))