Average Error: 21.7 → 0.1
Time: 1.9s
Precision: binary64
\[\sqrt{x \cdot x + y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.787085724672916 \cdot 10^{+148}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(0.125, \frac{y}{x \cdot x}, -0.5\right)}{x} - x\\ \mathbf{elif}\;x \leq 1.1157196809697832 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.787085724672916 \cdot 10^{+148}:\\
\;\;\;\;\frac{y \cdot \mathsf{fma}\left(0.125, \frac{y}{x \cdot x}, -0.5\right)}{x} - x\\

\mathbf{elif}\;x \leq 1.1157196809697832 \cdot 10^{+125}:\\
\;\;\;\;\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 -3.787085724672916e+148)
   (- (/ (* y (fma 0.125 (/ y (* x x)) -0.5)) x) x)
   (if (<= x 1.1157196809697832e+125) (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 <= -3.787085724672916e+148) {
		tmp = ((y * fma(0.125, (y / (x * x)), -0.5)) / x) - x;
	} else if (x <= 1.1157196809697832e+125) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.7
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.7870857246729161e148

    1. Initial program 61.7

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

    2. Simplified61.7

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

    3. Taylor expanded in x around -inf 15.6

      \[\leadsto \color{blue}{0.125 \cdot \frac{{y}^{2}}{{x}^{3}} - \left(0.5 \cdot \frac{y}{x} + x\right)} \]

    4. Simplified0.1

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\frac{y}{x} \cdot \frac{0.125}{x} + -0.5\right) - x} \]

    5. Applied associate-*l/_binary640.1

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{y}{x} \cdot \frac{0.125}{x} + -0.5\right)}{x}} - x \]

    6. Simplified0.1

      \[\leadsto \frac{\color{blue}{y \cdot \mathsf{fma}\left(0.125, \frac{y}{x \cdot x}, -0.5\right)}}{x} - x \]

    if -3.7870857246729161e148 < x < 1.1157196809697832e125

    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 1.1157196809697832e125 < x

    1. Initial program 54.8

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

    2. Simplified54.8

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

    3. Taylor expanded in x around inf 0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.787085724672916 \cdot 10^{+148}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(0.125, \frac{y}{x \cdot x}, -0.5\right)}{x} - x\\ \mathbf{elif}\;x \leq 1.1157196809697832 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))