Average Error: 23.5 → 0.8
Time: 19.4s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.0360529674537215 \cdot 10^{-253}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.0360529674537215 \cdot 10^{-253}:\\
\;\;\;\;\left(-x\right) \cdot y\\

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

\end{array}
double f(double x, double y, double z) {
        double r45590358 = x;
        double r45590359 = y;
        double r45590360 = r45590359 * r45590359;
        double r45590361 = z;
        double r45590362 = r45590361 * r45590361;
        double r45590363 = r45590360 - r45590362;
        double r45590364 = sqrt(r45590363);
        double r45590365 = r45590358 * r45590364;
        return r45590365;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r45590366 = y;
        double r45590367 = -3.0360529674537215e-253;
        bool r45590368 = r45590366 <= r45590367;
        double r45590369 = x;
        double r45590370 = -r45590369;
        double r45590371 = r45590370 * r45590366;
        double r45590372 = r45590369 * r45590366;
        double r45590373 = r45590368 ? r45590371 : r45590372;
        return r45590373;
}

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

Original23.5
Target0.5
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.0360529674537215e-253

    1. Initial program 23.3

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

    2. Taylor expanded around -inf 0.5

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

    3. Simplified0.5

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

    if -3.0360529674537215e-253 < y

    1. Initial program 23.7

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.0360529674537215 \cdot 10^{-253}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"

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

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