Average Error: 38.3 → 25.3
Time: 2.5s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3228887431998753 \cdot 10^{+134}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2830528115639911 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3228887431998753 \cdot 10^{+134}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 1.2830528115639911 \cdot 10^{+128}:\\
\;\;\;\;\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 -1.3228887431998753e+134)
   (- x)
   (if (<= x 1.2830528115639911e+128)
     (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 <= -1.3228887431998753e+134) {
		tmp = -x;
	} else if (x <= 1.2830528115639911e+128) {
		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

Original38.3
Target25.8
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.32288874319987533e134

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 14.7

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

    3. Simplified14.7

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

    if -1.32288874319987533e134 < x < 1.28305281156399112e128

    1. Initial program 29.5

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

    if 1.28305281156399112e128 < x

    1. Initial program 59.5

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

    2. Taylor expanded around inf 15.7

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

  4. Final simplification25.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3228887431998753 \cdot 10^{+134}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2830528115639911 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))