Average Error: 24.8 → 0.9
Time: 11.0s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.650359340362325641012254610154623215362 \cdot 10^{-234}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.650359340362325641012254610154623215362 \cdot 10^{-234}:\\
\;\;\;\;-x \cdot y\\

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

\end{array}
double f(double x, double y, double z) {
        double r551395 = x;
        double r551396 = y;
        double r551397 = r551396 * r551396;
        double r551398 = z;
        double r551399 = r551398 * r551398;
        double r551400 = r551397 - r551399;
        double r551401 = sqrt(r551400);
        double r551402 = r551395 * r551401;
        return r551402;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r551403 = y;
        double r551404 = -1.6503593403623256e-234;
        bool r551405 = r551403 <= r551404;
        double r551406 = x;
        double r551407 = r551406 * r551403;
        double r551408 = -r551407;
        double r551409 = r551405 ? r551408 : r551407;
        return r551409;
}

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

Original24.8
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt 2.581609648825169508994985860317034908583 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.6503593403623256e-234

    1. Initial program 24.9

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

    2. Taylor expanded around -inf 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{-x \cdot y}\]

    if -1.6503593403623256e-234 < y

    1. Initial program 24.7

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

    2. Taylor expanded around inf 1.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.650359340362325641012254610154623215362 \cdot 10^{-234}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))))