Average Error: 21.7 → 0.1
Time: 11.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\

\end{array}
double f(double x, double y) {
        double r373084 = x;
        double r373085 = r373084 * r373084;
        double r373086 = y;
        double r373087 = r373085 + r373086;
        double r373088 = sqrt(r373087);
        return r373088;
}
↓
double f(double x, double y) {
        double r373089 = x;
        double r373090 = -1.2996772002233855e+154;
        bool r373091 = r373089 <= r373090;
        double r373092 = y;
        double r373093 = r373092 / r373089;
        double r373094 = -0.5;
        double r373095 = r373093 * r373094;
        double r373096 = r373095 - r373089;
        double r373097 = 7.419817825242465e+111;
        bool r373098 = r373089 <= r373097;
        double r373099 = r373089 * r373089;
        double r373100 = r373099 + r373092;
        double r373101 = sqrt(r373100);
        double r373102 = 0.5;
        double r373103 = r373102 * r373093;
        double r373104 = r373089 + r373103;
        double r373105 = r373098 ? r373101 : r373104;
        double r373106 = r373091 ? r373096 : r373105;
        return r373106;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.7
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2996772002233855e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    3. Simplified0

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{-1}{2} - x}\]

    if -1.2996772002233855e+154 < x < 7.419817825242465e+111

    1. Initial program 0.0

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

    if 7.419817825242465e+111 < x

    1. Initial program 50.8

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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