Average Error: 25.0 → 0.7
Time: 24.9s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.814361563492097218084667841876457735263 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.814361563492097218084667841876457735263 \cdot 10^{-270}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot y\\

\end{array}
double f(double x, double y, double z) {
        double r486800 = x;
        double r486801 = y;
        double r486802 = r486801 * r486801;
        double r486803 = z;
        double r486804 = r486803 * r486803;
        double r486805 = r486802 - r486804;
        double r486806 = sqrt(r486805);
        double r486807 = r486800 * r486806;
        return r486807;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r486808 = y;
        double r486809 = -1.8143615634920972e-270;
        bool r486810 = r486808 <= r486809;
        double r486811 = x;
        double r486812 = -r486808;
        double r486813 = r486811 * r486812;
        double r486814 = r486811 * r486808;
        double r486815 = r486810 ? r486813 : r486814;
        return r486815;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.0
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt 2.581609648825169508994985860317034908583 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.8143615634920972e-270

    1. Initial program 24.9

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

    2. Taylor expanded around -inf 0.6

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

    3. Simplified0.6

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

    if -1.8143615634920972e-270 < y

    1. Initial program 25.0

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

    2. Taylor expanded around inf 0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.814361563492097218084667841876457735263 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))