Average Error: 24.5 → 0.6
Time: 4.4s
Precision: binary64
Cost: 13697
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 7.43280876674966 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{y + z}\right) \cdot \sqrt{y - z}\\ \end{array}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 7.43280876674966 \cdot 10^{-289}:\\
\;\;\;\;x \cdot \left(-y\right)\\

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

\end{array}
(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y 7.43280876674966e-289)
   (* 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 <= 7.43280876674966e-289) {
		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

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

Alternatives

Alternative 1
Error0.7
Cost577
\[\begin{array}{l} \mathbf{if}\;y \leq 3.0009870283414602 \cdot 10^{-291}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]
Alternative 2
Error30.8
Cost192
\[y \cdot x\]
Alternative 3
Error58.7
Cost64
\[0\]
Alternative 4
Error61.7
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if y < 7.43280876674966017e-289

    1. Initial program 24.1

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

    2. Taylor expanded around -inf 0.8

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

    3. Simplified0.8

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

    4. Simplified0.8

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

    if 7.43280876674966017e-289 < y

    1. Initial program 24.9

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

    3. Applied difference-of-squares_binary64_1880224.9

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

    4. Applied sqrt-prod_binary64_188490.4

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

    5. Applied associate-*r*_binary64_187730.5

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

    6. Simplified0.5

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

    7. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.43280876674966 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{y + z}\right) \cdot \sqrt{y - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))