Average Error: 24.9 → 0.5
Time: 3.3s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 2.206379454757968 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 2.206379454757968 \cdot 10^{-302}:\\
\;\;\;\;x \cdot \left(-y\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\


\end{array}

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.206379454757968e-302)
   (* x (- y))
   (* x (* (sqrt (+ y z)) (sqrt (- y z))))))
double code(double x, double y, double z) {
	return x * sqrt((y * y) - (z * z));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.206379454757968e-302) {
		tmp = x * -y;
	} else {
		tmp = x * (sqrt(y + z) * sqrt(y - z));
	}
	return tmp;
}

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

Original24.9
Target0.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < 2.206379454757968e-302

    1. Initial program 24.3

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

    2. Taylor expanded in y around -inf 0.6

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

    3. Simplified0.6

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

    if 2.206379454757968e-302 < y

    1. Initial program 25.5

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

    2. Applied difference-of-squares_binary6425.5

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

    3. Applied sqrt-prod_binary640.4

      \[\leadsto x \cdot \color{blue}{\left(\sqrt{y + z} \cdot \sqrt{y - z}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.206379454757968 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

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