Average Error: 25.1 → 0.5
Time: 6.3s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -8.240555728114612 \cdot 10^{-308}:\\ \;\;\;\;-y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y - z} \cdot \sqrt{y + z}\right)\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -8.240555728114612 \cdot 10^{-308}:\\
\;\;\;\;-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 -8.240555728114612e-308)
   (- (* 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 <= -8.240555728114612e-308) {
		tmp = -(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

Original25.1
Target0.5
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 < -8.2405557281146121e-308

    1. Initial program 25.5

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

    2. Taylor expanded around -inf 0.5

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

    if -8.2405557281146121e-308 < y

    1. Initial program 24.8

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]
    2. Using strategy rm

    3. Applied difference-of-squares_binary64_1436924.8

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

    4. Applied sqrt-prod_binary64_144160.4

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

    5. Simplified0.4

      \[\leadsto x \cdot \left(\color{blue}{\sqrt{z + y}} \cdot \sqrt{y - z}\right)\]
    6. Using strategy rm

    7. Applied *-commutative_binary64_143310.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.240555728114612 \cdot 10^{-308}:\\ \;\;\;\;-y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y - z} \cdot \sqrt{y + z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021076 
(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)))))