Average Error: 21.4 → 21.4
Time: 1.1s
Precision: binary64
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}
double code(double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) sqrt(((double) (1.0 + ((double) (x * x))))))))))))));
}
↓
double code(double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) sqrt(((double) (1.0 + ((double) (x * x))))))))))))));
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 21.4

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}\]

  2. Final simplification21.4

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "(sqrt (* 0.5 (+ 1 (/ x (sqrt (+ 1 (* x x)))))))"
  :precision binary64
  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ 1.0 (* x x))))))))