Average Error: 25.5 → 0.9
Time: 5.5s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -4.957508820182921 \cdot 10^{-247}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -4.957508820182921 \cdot 10^{-247}:\\
\;\;\;\;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 -4.957508820182921e-247)
   (* 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 <= -4.957508820182921e-247) {
		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

Original25.5
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.957508820182921e-247

    1. Initial program 25.4

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

    2. Taylor expanded in y 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 -4.957508820182921e-247 < y

    1. Initial program 25.5

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

    2. Applied egg-rr1.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.957508820182921 \cdot 10^{-247}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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)))))