Average Error: 25.2 → 0.7
Time: 9.5s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.65053048987856224 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.65053048987856224 \cdot 10^{-263}:\\
\;\;\;\;x \cdot \left(-1 \cdot y\right)\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (x * ((double) sqrt(((double) (((double) (y * y)) - ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((y <= -5.650530489878562e-263)) {
		VAR = ((double) (x * ((double) (-1.0 * y))));
	} else {
		VAR = ((double) (x * y));
	}
	return VAR;
}

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

Original25.2
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt 2.58160964882516951 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.65053048987856224e-263

    1. Initial program 25.1

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

    2. Taylor expanded around -inf 0.6

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

    if -5.65053048987856224e-263 < y

    1. Initial program 25.2

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

    2. Taylor expanded around inf 0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.65053048987856224 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \left(-1 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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) (neg (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

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