Average Error: 37.9 → 25.7
Time: 14.3s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.279231778552234340489453953388016181689 \cdot 10^{128}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 2.128703798252542087511374205356575227021 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.279231778552234340489453953388016181689 \cdot 10^{128}:\\
\;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\mathbf{elif}\;x \le 2.128703798252542087511374205356575227021 \cdot 10^{74}:\\
\;\;\;\;\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 r1007607 = x;
        double r1007608 = r1007607 * r1007607;
        double r1007609 = y;
        double r1007610 = r1007609 * r1007609;
        double r1007611 = r1007608 + r1007610;
        double r1007612 = z;
        double r1007613 = r1007612 * r1007612;
        double r1007614 = r1007611 + r1007613;
        double r1007615 = 3.0;
        double r1007616 = r1007614 / r1007615;
        double r1007617 = sqrt(r1007616);
        return r1007617;
}
↓
double f(double x, double y, double z) {
        double r1007618 = x;
        double r1007619 = -1.2792317785522343e+128;
        bool r1007620 = r1007618 <= r1007619;
        double r1007621 = 0.3333333333333333;
        double r1007622 = sqrt(r1007621);
        double r1007623 = r1007618 * r1007622;
        double r1007624 = -r1007623;
        double r1007625 = 2.128703798252542e+74;
        bool r1007626 = r1007618 <= r1007625;
        double r1007627 = z;
        double r1007628 = y;
        double r1007629 = 2.0;
        double r1007630 = pow(r1007628, r1007629);
        double r1007631 = fma(r1007618, r1007618, r1007630);
        double r1007632 = fma(r1007627, r1007627, r1007631);
        double r1007633 = sqrt(r1007632);
        double r1007634 = 3.0;
        double r1007635 = sqrt(r1007634);
        double r1007636 = r1007633 / r1007635;
        double r1007637 = r1007626 ? r1007636 : r1007623;
        double r1007638 = r1007620 ? r1007624 : r1007637;
        return r1007638;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original37.9
Target25.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2792317785522343e+128

    1. Initial program 59.3

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

    2. Taylor expanded around -inf 16.3

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

    3. Simplified16.3

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

    if -1.2792317785522343e+128 < x < 2.128703798252542e+74

    1. Initial program 29.2

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

    3. Applied sqrt-div29.3

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

    4. Simplified29.3

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

    if 2.128703798252542e+74 < x

    1. Initial program 52.2

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

    2. Taylor expanded around inf 20.0

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.279231778552234340489453953388016181689 \cdot 10^{128}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 2.128703798252542087511374205356575227021 \cdot 10^{74}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, {y}^{2}\right)\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))