Average Error: 21.3 → 0.2
Time: 1.2s
Precision: binary64
\[\sqrt{x \cdot x + y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0520697127366559 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 8.089445965002033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0520697127366559 \cdot 10^{+154}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 8.089445965002033 \cdot 10^{+96}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

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


\end{array}

(FPCore (x y) :precision binary64 (sqrt (+ (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0520697127366559e+154)
   (- x)
   (if (<= x 8.089445965002033e+96) (sqrt (fma x x y)) x)))
double code(double x, double y) {
	return sqrt((x * x) + y);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -1.0520697127366559e+154) {
		tmp = -x;
	} else if (x <= 8.089445965002033e+96) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.3
Target0.6
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.05206971273665588e154

    1. Initial program 64.0

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

    2. Simplified64.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]

    3. Taylor expanded in x around -inf 0.0

      \[\leadsto \color{blue}{-1 \cdot x} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{-x} \]

    if -1.05206971273665588e154 < x < 8.08944596500203324e96

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]

    if 8.08944596500203324e96 < x

    1. Initial program 48.0

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

    2. Simplified48.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]

    3. Taylor expanded in x around inf 0.8

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0520697127366559 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 8.089445965002033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))