(if (< x 1/10000) (+ 1 (* 1/2 x)) (sqrt (+ 1 x)))

Average Error: 0.0 → 0.0

Time: 787.0ms

Precision: binary64

Math

\[\begin{array}{l} \mathbf{if}\;x \lt 1.00000000000000005 \cdot 10^{-4}:\\ \;\;\;\;1 + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \lt 1.00000000000000005 \cdot 10^{-4}:\\ \;\;\;\;1 + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x}\\ \end{array}\]

C

double code(double x) {
	double VAR;
	if ((x < 0.0001)) {
		VAR = ((double) (1.0 + ((double) (0.5 * x))));
	} else {
		VAR = ((double) sqrt(((double) (1.0 + x))));
	}
	return VAR;
}

↓

double code(double x) {
	double VAR;
	if ((x < 0.0001)) {
		VAR = ((double) (1.0 + ((double) (0.5 * x))));
	} else {
		VAR = ((double) sqrt(((double) (1.0 + x))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[\begin{array}{l} \mathbf{if}\;x \lt 1.00000000000000005 \cdot 10^{-4}:\\ \;\;\;\;1 + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x}\\ \end{array}\]

2. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \lt 1.00000000000000005 \cdot 10^{-4}:\\ \;\;\;\;1 + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "(if (< x 1/10000) (+ 1 (* 1/2 x)) (sqrt (+ 1 x)))"
  :precision binary64
  (if (< x 0.0001) (+ 1.0 (* 0.5 x)) (sqrt (+ 1.0 x))))