Average Error: 21.5 → 0.0
Time: 8.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.391701187709677107962775681634160419303 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.391701187709677107962775681634160419303 \cdot 10^{153}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r24913102 = x;
        double r24913103 = r24913102 * r24913102;
        double r24913104 = y;
        double r24913105 = r24913103 + r24913104;
        double r24913106 = sqrt(r24913105);
        return r24913106;
}
↓
double f(double x, double y) {
        double r24913107 = x;
        double r24913108 = -9.391701187709677e+153;
        bool r24913109 = r24913107 <= r24913108;
        double r24913110 = -0.5;
        double r24913111 = y;
        double r24913112 = r24913107 / r24913111;
        double r24913113 = r24913110 / r24913112;
        double r24913114 = r24913113 - r24913107;
        double r24913115 = 1.7202383854095123e+146;
        bool r24913116 = r24913107 <= r24913115;
        double r24913117 = r24913107 * r24913107;
        double r24913118 = r24913117 + r24913111;
        double r24913119 = sqrt(r24913118);
        double r24913120 = r24913111 / r24913107;
        double r24913121 = 0.5;
        double r24913122 = r24913120 * r24913121;
        double r24913123 = r24913122 + r24913107;
        double r24913124 = r24913116 ? r24913119 : r24913123;
        double r24913125 = r24913109 ? r24913114 : r24913124;
        return r24913125;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.5
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -9.391701187709677e+153

    1. Initial program 63.9

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

    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 -9.391701187709677e+153 < x < 1.7202383854095123e+146

    1. Initial program 0.0

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

    if 1.7202383854095123e+146 < x

    1. Initial program 61.3

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

    2. Taylor expanded around inf 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.391701187709677107962775681634160419303 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))