Average Error: 20.7 → 1.5
Time: 1.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3390824056870139 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 4.608597005595938 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3390824056870139 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 4.608597005595938 \cdot 10^{-20}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r454644 = x;
        double r454645 = r454644 * r454644;
        double r454646 = y;
        double r454647 = r454645 + r454646;
        double r454648 = sqrt(r454647);
        return r454648;
}
↓
double f(double x, double y) {
        double r454649 = x;
        double r454650 = -1.3390824056870139e+154;
        bool r454651 = r454649 <= r454650;
        double r454652 = 0.5;
        double r454653 = y;
        double r454654 = r454653 / r454649;
        double r454655 = r454652 * r454654;
        double r454656 = r454649 + r454655;
        double r454657 = -r454656;
        double r454658 = 4.6085970055959376e-20;
        bool r454659 = r454649 <= r454658;
        double r454660 = r454649 * r454649;
        double r454661 = r454660 + r454653;
        double r454662 = sqrt(r454661);
        double r454663 = r454659 ? r454662 : r454656;
        double r454664 = r454651 ? r454657 : r454663;
        return r454664;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.7
Target0.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3390824056870139e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.1

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

    if -1.3390824056870139e+154 < x < 4.6085970055959376e-20

    1. Initial program 0.0

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

    if 4.6085970055959376e-20 < x

    1. Initial program 31.9

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

    2. Taylor expanded around inf 4.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3390824056870139 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 4.608597005595938 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))