Average Error: 22.0 → 0.1
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r456053 = x;
        double r456054 = r456053 * r456053;
        double r456055 = y;
        double r456056 = r456054 + r456055;
        double r456057 = sqrt(r456056);
        return r456057;
}
↓
double f(double x, double y) {
        double r456058 = x;
        double r456059 = -1.3352056197781018e+154;
        bool r456060 = r456058 <= r456059;
        double r456061 = 0.5;
        double r456062 = y;
        double r456063 = r456062 / r456058;
        double r456064 = r456061 * r456063;
        double r456065 = r456058 + r456064;
        double r456066 = -r456065;
        double r456067 = 8.456870552497892e+117;
        bool r456068 = r456058 <= r456067;
        double r456069 = r456058 * r456058;
        double r456070 = r456069 + r456062;
        double r456071 = sqrt(r456070);
        double r456072 = r456068 ? r456071 : r456065;
        double r456073 = r456060 ? r456066 : r456072;
        return r456073;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.6
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3352056197781018e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.3352056197781018e+154 < x < 8.456870552497892e+117

    1. Initial program 0.0

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

    if 8.456870552497892e+117 < x

    1. Initial program 52.8

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

    2. Taylor expanded around inf 0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))