Average Error: 23.5 → 0.8
Time: 16.3s
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 r33645162 = x;
        double r33645163 = y;
        double r33645164 = r33645163 * r33645163;
        double r33645165 = z;
        double r33645166 = r33645165 * r33645165;
        double r33645167 = r33645164 - r33645166;
        double r33645168 = sqrt(r33645167);
        double r33645169 = r33645162 * r33645168;
        return r33645169;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r33645170 = y;
        double r33645171 = -3.0360529674537215e-253;
        bool r33645172 = r33645170 <= r33645171;
        double r33645173 = x;
        double r33645174 = -r33645173;
        double r33645175 = r33645174 * r33645170;
        double r33645176 = r33645173 * r33645170;
        double r33645177 = r33645172 ? r33645175 : r33645176;
        return r33645177;
}

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. Using strategy rm

    3. Applied difference-of-squares23.2

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

    4. Taylor expanded around -inf 0.5

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

    5. Simplified0.5

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

    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)))))