Average Error: 38.1 → 26.4
Time: 4.4s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.6747749261665936 \cdot 10^{+83}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.8403577741388615 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.6747749261665936 \cdot 10^{+83}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 2.8403577741388615 \cdot 10^{+85}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6747749261665936e+83)
   (- x)
   (if (<= x 2.8403577741388615e+85)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     x)))
double code(double x, double y, double z) {
	return sqrt(((x * x) + (y * y)) + (z * z));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6747749261665936e+83) {
		tmp = -x;
	} else if (x <= 2.8403577741388615e+85) {
		tmp = sqrt(((x * x) + (y * y)) + (z * z));
	} else {
		tmp = x;
	}
	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

Original38.1
Target25.1
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.6747749261665936e83

    1. Initial program 52.8

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Taylor expanded around -inf 20.8

      \[\leadsto \color{blue}{-1 \cdot x}\]

    3. Simplified20.8

      \[\leadsto \color{blue}{-x}\]

    if -1.6747749261665936e83 < x < 2.84035777413886155e85

    1. Initial program 30.1

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    if 2.84035777413886155e85 < x

    1. Initial program 52.4

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Taylor expanded around inf 18.7

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6747749261665936 \cdot 10^{+83}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.8403577741388615 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))