Average Error: 21.1 → 0.2
Time: 14.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r25345693 = x;
        double r25345694 = r25345693 * r25345693;
        double r25345695 = y;
        double r25345696 = r25345694 + r25345695;
        double r25345697 = sqrt(r25345696);
        return r25345697;
}
↓
double f(double x, double y) {
        double r25345698 = x;
        double r25345699 = -1.3584562935631267e+154;
        bool r25345700 = r25345698 <= r25345699;
        double r25345701 = -0.5;
        double r25345702 = y;
        double r25345703 = r25345698 / r25345702;
        double r25345704 = r25345701 / r25345703;
        double r25345705 = r25345704 - r25345698;
        double r25345706 = 2.2470376765740678e+83;
        bool r25345707 = r25345698 <= r25345706;
        double r25345708 = r25345698 * r25345698;
        double r25345709 = r25345708 + r25345702;
        double r25345710 = sqrt(r25345709);
        double r25345711 = r25345702 / r25345698;
        double r25345712 = 0.5;
        double r25345713 = r25345711 * r25345712;
        double r25345714 = r25345713 + r25345698;
        double r25345715 = r25345707 ? r25345710 : r25345714;
        double r25345716 = r25345700 ? r25345705 : r25345715;
        return r25345716;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
Target0.4
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 < -1.3584562935631267e+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

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

    if -1.3584562935631267e+154 < x < 2.2470376765740678e+83

    1. Initial program 0.0

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

    if 2.2470376765740678e+83 < x

    1. Initial program 43.9

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))