Average Error: 21.8 → 0.4
Time: 2.6s
Precision: binary64
\[\sqrt{x \cdot x + y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -7.825380784639379 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.4172583188134197 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \end{array} \]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -7.825380784639379 \cdot 10^{+153}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 1.4172583188134197 \cdot 10^{+67}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\


\end{array}

(FPCore (x y) :precision binary64 (sqrt (+ (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -7.825380784639379e+153)
   (- x)
   (if (<= x 1.4172583188134197e+67) (sqrt (fma x x y)) (fma 0.5 (/ y x) x))))
double code(double x, double y) {
	return sqrt((x * x) + y);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -7.825380784639379e+153) {
		tmp = -x;
	} else if (x <= 1.4172583188134197e+67) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = fma(0.5, (y / x), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.8
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -7.8253807846393787e153

    1. Initial program 63.9

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

    2. Simplified63.9

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

    3. Taylor expanded in x around -inf 0.0

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

    4. Simplified0.0

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

    if -7.8253807846393787e153 < x < 1.4172583188134197e67

    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 1.4172583188134197e67 < x

    1. Initial program 42.7

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

    2. Simplified42.7

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

    3. Taylor expanded in x around inf 1.5

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

    4. Simplified1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.825380784639379 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.4172583188134197 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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)))