Average Error: 24.9 → 1.2
Time: 3.4s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 3.382080778465714 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \frac{z \cdot \left(z \cdot x\right)}{y} - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 3.382080778465714 \cdot 10^{-303}:\\
\;\;\;\;0.5 \cdot \frac{z \cdot \left(z \cdot x\right)}{y} - y \cdot x\\

\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 3.382080778465714e-303)
   (- (* 0.5 (/ (* z (* z x)) y)) (* y x))
   (* 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 <= 3.382080778465714e-303) {
		tmp = (0.5 * ((z * (z * x)) / y)) - (y * x);
	} 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
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < 3.38208077846571406e-303

    1. Initial program 24.5

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

    2. Taylor expanded in y around -inf 3.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} - y \cdot x} \]

    3. Applied unpow2_binary643.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(z \cdot z\right)} \cdot x}{y} - y \cdot x \]

    4. Applied associate-*l*_binary641.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{z \cdot \left(z \cdot x\right)}}{y} - y \cdot x \]

    if 3.38208077846571406e-303 < y

    1. Initial program 25.3

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

    2. Applied difference-of-squares_binary6425.3

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

    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 simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.382080778465714 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \frac{z \cdot \left(z \cdot x\right)}{y} - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))))