Average Error: 20.8 → 0.5
Time: 2.8s
Precision: binary64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3506077002236376 \cdot 10^{+154}:\\ \;\;\;\;-\left(x + 0.5 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 5.507042569157648 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3506077002236376 \cdot 10^{+154}:\\
\;\;\;\;-\left(x + 0.5 \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \leq 5.507042569157648 \cdot 10^{+57}:\\
\;\;\;\;\sqrt{y + x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.3506077002236376e+154)
   (- (+ x (* 0.5 (/ y x))))
   (if (<= x 5.507042569157648e+57) (sqrt (+ y (* x x))) x)))
double code(double x, double y) {
	return sqrt((x * x) + y);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -1.3506077002236376e+154) {
		tmp = -(x + (0.5 * (y / x)));
	} else if (x <= 5.507042569157648e+57) {
		tmp = sqrt(y + (x * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3506077002236376e154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    if -1.3506077002236376e154 < x < 5.5070425691576483e57

    1. Initial program 0.0

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

    if 5.5070425691576483e57 < x

    1. Initial program 39.5

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

    2. Taylor expanded around inf 2.0

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3506077002236376 \cdot 10^{+154}:\\ \;\;\;\;-\left(x + 0.5 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 5.507042569157648 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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)))