Average Error: 24.8 → 0.9
Time: 13.9s
Precision: 64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.650359340362325641012254610154623215362 \cdot 10^{-234}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.650359340362325641012254610154623215362 \cdot 10^{-234}:\\
\;\;\;\;\left(-x\right) \cdot y\\

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

\end{array}
double f(double x, double y, double z) {
        double r26589340 = x;
        double r26589341 = y;
        double r26589342 = r26589341 * r26589341;
        double r26589343 = z;
        double r26589344 = r26589343 * r26589343;
        double r26589345 = r26589342 - r26589344;
        double r26589346 = sqrt(r26589345);
        double r26589347 = r26589340 * r26589346;
        return r26589347;
}
↓
double f(double x, double y, double __attribute__((unused)) z) {
        double r26589348 = y;
        double r26589349 = -1.6503593403623256e-234;
        bool r26589350 = r26589348 <= r26589349;
        double r26589351 = x;
        double r26589352 = -r26589351;
        double r26589353 = r26589352 * r26589348;
        double r26589354 = r26589351 * r26589348;
        double r26589355 = r26589350 ? r26589353 : r26589354;
        return r26589355;
}

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

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

    3. Simplified0.5

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

    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

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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)))))