Average Error: 21.4 → 0.2
Time: 25.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.344192746470904425870702492238984808211 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{\frac{x}{\frac{-1}{2}}} - x\\ \mathbf{elif}\;x \le 3.146509911612528293591568472384734910948 \cdot 10^{98}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.344192746470904425870702492238984808211 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{\frac{x}{\frac{-1}{2}}} - x\\

\mathbf{elif}\;x \le 3.146509911612528293591568472384734910948 \cdot 10^{98}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r23917558 = x;
        double r23917559 = r23917558 * r23917558;
        double r23917560 = y;
        double r23917561 = r23917559 + r23917560;
        double r23917562 = sqrt(r23917561);
        return r23917562;
}
↓
double f(double x, double y) {
        double r23917563 = x;
        double r23917564 = -1.3441927464709044e+154;
        bool r23917565 = r23917563 <= r23917564;
        double r23917566 = y;
        double r23917567 = -0.5;
        double r23917568 = r23917563 / r23917567;
        double r23917569 = r23917566 / r23917568;
        double r23917570 = r23917569 - r23917563;
        double r23917571 = 3.146509911612528e+98;
        bool r23917572 = r23917563 <= r23917571;
        double r23917573 = r23917563 * r23917563;
        double r23917574 = r23917573 + r23917566;
        double r23917575 = sqrt(r23917574);
        double r23917576 = 0.5;
        double r23917577 = r23917566 / r23917563;
        double r23917578 = r23917576 * r23917577;
        double r23917579 = r23917578 + r23917563;
        double r23917580 = r23917572 ? r23917575 : r23917579;
        double r23917581 = r23917565 ? r23917570 : r23917580;
        return r23917581;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.4
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3441927464709044e+154

    1. Initial program 64.0

      \[\sqrt{x \cdot x + y}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt64.0

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{x \cdot x + y} \cdot \sqrt[3]{x \cdot x + y}\right) \cdot \sqrt[3]{x \cdot x + y}}}\]

    4. Taylor expanded around -inf 0

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

    5. Simplified0

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

    if -1.3441927464709044e+154 < x < 3.146509911612528e+98

    1. Initial program 0.0

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

    if 3.146509911612528e+98 < x

    1. Initial program 47.4

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

    2. Taylor expanded around inf 0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.344192746470904425870702492238984808211 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{\frac{x}{\frac{-1}{2}}} - x\\ \mathbf{elif}\;x \le 3.146509911612528293591568472384734910948 \cdot 10^{98}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))