Average Error: 22.0 → 0.3
Time: 1.6s
Precision: binary64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3315812935511254 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 1.7236320068184444 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3315812935511254 \cdot 10^{+154}:\\
\;\;\;\;y \cdot \frac{-0.5}{x} - x\\

\mathbf{elif}\;x \leq 1.7236320068184444 \cdot 10^{+81}:\\
\;\;\;\;\sqrt{y + x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{0.5}{x}\\

\end{array}
double code(double x, double y) {
	return ((double) sqrt(((double) (((double) (x * x)) + y))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((x <= -1.3315812935511254e+154)) {
		VAR = ((double) (((double) (y * (-0.5 / x))) - x));
	} else {
		double VAR_1;
		if ((x <= 1.7236320068184444e+81)) {
			VAR_1 = ((double) sqrt(((double) (y + ((double) (x * x))))));
		} else {
			VAR_1 = ((double) (x + ((double) (y * (0.5 / x)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.33158129355112536e154

    1. Initial program Error: 64.0 bits

      \[\sqrt{x \cdot x + y}\]

    2. Taylor expanded around -inf Error: 0 bits

      \[\leadsto \color{blue}{-\left(x + 0.5 \cdot \frac{y}{x}\right)}\]

    3. SimplifiedError: 0 bits

      \[\leadsto \color{blue}{y \cdot \frac{-0.5}{x} - x}\]

    if -1.33158129355112536e154 < x < 1.72363200681844438e81

    1. Initial program Error: 0.0 bits

      \[\sqrt{x \cdot x + y}\]

    if 1.72363200681844438e81 < x

    1. Initial program Error: 45.0 bits

      \[\sqrt{x \cdot x + y}\]

    2. Taylor expanded around inf Error: 1.1 bits

      \[\leadsto \color{blue}{x + 0.5 \cdot \frac{y}{x}}\]

    3. SimplifiedError: 1.1 bits

      \[\leadsto \color{blue}{x + y \cdot \frac{0.5}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 0.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3315812935511254 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 1.7236320068184444 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))