Average Error: 37.8 → 25.8
Time: 3.5s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.634413777114509 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 4.949626136573165 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.634413777114509 \cdot 10^{+97}:\\
\;\;\;\;\frac{-x}{\sqrt{3}}\\

\mathbf{elif}\;x \leq 4.949626136573165 \cdot 10^{+140}:\\
\;\;\;\;\sqrt{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.3333333333333333}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\sqrt{3}}\\

\end{array}
(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.634413777114509e+97)
   (/ (- x) (sqrt 3.0))
   (if (<= x 4.949626136573165e+140)
     (sqrt (* (+ (* x x) (+ (* y y) (* z z))) 0.3333333333333333))
     (/ x (sqrt 3.0)))))
double code(double x, double y, double z) {
	return ((double) sqrt((((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.634413777114509e+97)) {
		tmp = (((double) -(x)) / ((double) sqrt(3.0)));
	} else {
		double tmp_1;
		if ((x <= 4.949626136573165e+140)) {
			tmp_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (((double) (y * y)) + ((double) (z * z)))))) * 0.3333333333333333))));
		} else {
			tmp_1 = (x / ((double) sqrt(3.0)));
		}
		tmp = tmp_1;
	}
	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

Original37.8
Target25.5
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.634413777114509e97

    1. Initial program Error: 54.5 bits

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
    2. Using strategy rm

    3. Applied sqrt-divError: 54.6 bits

      \[\leadsto \color{blue}{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}}\]

    4. SimplifiedError: 54.6 bits

      \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}}{\sqrt{3}}\]

    5. Taylor expanded around -inf Error: 18.7 bits

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\sqrt{3}}\]

    6. SimplifiedError: 18.7 bits

      \[\leadsto \frac{\color{blue}{-x}}{\sqrt{3}}\]

    if -1.634413777114509e97 < x < 4.9496261365731651e140

    1. Initial program Error: 29.3 bits

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

    2. Taylor expanded around 0 Error: 29.3 bits

      \[\leadsto \sqrt{\color{blue}{0.3333333333333333 \cdot {x}^{2} + \left(0.3333333333333333 \cdot {y}^{2} + 0.3333333333333333 \cdot {z}^{2}\right)}}\]

    3. SimplifiedError: 29.3 bits

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.3333333333333333}}\]

    if 4.9496261365731651e140 < x

    1. Initial program Error: 62.1 bits

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
    2. Using strategy rm

    3. Applied sqrt-divError: 62.1 bits

      \[\leadsto \color{blue}{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}}\]

    4. SimplifiedError: 62.1 bits

      \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}}{\sqrt{3}}\]

    5. Taylor expanded around inf Error: 16.2 bits

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

  4. Final simplificationError: 25.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.634413777114509 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 4.949626136573165 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(x \cdot x + \left(y \cdot y + z \cdot z\right)\right) \cdot 0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))