Average Error: 37.6 → 25.6
Time: 20.7s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.839580010133738883814333111888121327705 \cdot 10^{89}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 1.537188568549186890366688724633159259389 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.839580010133738883814333111888121327705 \cdot 10^{89}:\\
\;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\mathbf{elif}\;x \le 1.537188568549186890366688724633159259389 \cdot 10^{101}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}{\sqrt{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\end{array}
double f(double x, double y, double z) {
        double r569120 = x;
        double r569121 = r569120 * r569120;
        double r569122 = y;
        double r569123 = r569122 * r569122;
        double r569124 = r569121 + r569123;
        double r569125 = z;
        double r569126 = r569125 * r569125;
        double r569127 = r569124 + r569126;
        double r569128 = 3.0;
        double r569129 = r569127 / r569128;
        double r569130 = sqrt(r569129);
        return r569130;
}
↓
double f(double x, double y, double z) {
        double r569131 = x;
        double r569132 = -4.839580010133739e+89;
        bool r569133 = r569131 <= r569132;
        double r569134 = 0.3333333333333333;
        double r569135 = sqrt(r569134);
        double r569136 = r569131 * r569135;
        double r569137 = -r569136;
        double r569138 = 1.537188568549187e+101;
        bool r569139 = r569131 <= r569138;
        double r569140 = z;
        double r569141 = y;
        double r569142 = 2.0;
        double r569143 = pow(r569141, r569142);
        double r569144 = fma(r569131, r569131, r569143);
        double r569145 = fma(r569140, r569140, r569144);
        double r569146 = sqrt(r569145);
        double r569147 = 3.0;
        double r569148 = sqrt(r569147);
        double r569149 = r569146 / r569148;
        double r569150 = r569139 ? r569149 : r569136;
        double r569151 = r569133 ? r569137 : r569150;
        return r569151;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original37.6
Target25.5
Herbie25.6
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.839580010133739e+89

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 19.7

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.3333333333333333148296162562473909929395}\right)}\]

    3. Simplified19.7

      \[\leadsto \color{blue}{-x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]

    if -4.839580010133739e+89 < x < 1.537188568549187e+101

    1. Initial program 29.1

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

    3. Applied sqrt-div29.2

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

    4. Simplified29.2

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}}{\sqrt{3}}\]

    if 1.537188568549187e+101 < x

    1. Initial program 55.2

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

    2. Taylor expanded around inf 17.6

      \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.839580010133738883814333111888121327705 \cdot 10^{89}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 1.537188568549186890366688724633159259389 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3)))