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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.99047635474855928770919369247857947665 \cdot 10^{124}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 9.333719392464218207997872709553383250537 \cdot 10^{137}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -3.99047635474855928770919369247857947665 \cdot 10^{124}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\

\mathbf{elif}\;x \le 9.333719392464218207997872709553383250537 \cdot 10^{137}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

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

\end{array}

double f(double x, double y, double z) {
        double r912653 = x;
        double r912654 = r912653 * r912653;
        double r912655 = y;
        double r912656 = r912655 * r912655;
        double r912657 = r912654 + r912656;
        double r912658 = z;
        double r912659 = r912658 * r912658;
        double r912660 = r912657 + r912659;
        double r912661 = 3.0;
        double r912662 = r912660 / r912661;
        double r912663 = sqrt(r912662);
        return r912663;
}

↓

double f(double x, double y, double z) {
        double r912664 = x;
        double r912665 = -3.9904763547485593e+124;
        bool r912666 = r912664 <= r912665;
        double r912667 = 1.0;
        double r912668 = 3.0;
        double r912669 = cbrt(r912668);
        double r912670 = r912669 * r912669;
        double r912671 = r912667 / r912670;
        double r912672 = sqrt(r912671);
        double r912673 = -1.0;
        double r912674 = r912667 / r912669;
        double r912675 = sqrt(r912674);
        double r912676 = r912675 * r912664;
        double r912677 = r912673 * r912676;
        double r912678 = r912672 * r912677;
        double r912679 = 9.333719392464218e+137;
        bool r912680 = r912664 <= r912679;
        double r912681 = r912664 * r912664;
        double r912682 = y;
        double r912683 = r912682 * r912682;
        double r912684 = r912681 + r912683;
        double r912685 = z;
        double r912686 = r912685 * r912685;
        double r912687 = r912684 + r912686;
        double r912688 = r912687 / r912668;
        double r912689 = sqrt(r912688);
        double r912690 = 0.3333333333333333;
        double r912691 = sqrt(r912690);
        double r912692 = r912664 * r912691;
        double r912693 = r912680 ? r912689 : r912692;
        double r912694 = r912666 ? r912678 : r912693;
        return r912694;
}

Original	39.0
Target	26.3
Herbie	26.3

Derivation

Split input into 3 regimes
if x < -3.9904763547485593e+124
1. Initial program 58.5
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
2. Using strategy rm
3. Applied add-cube-cbrt58.5
  \[\leadsto \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}}\]
4. Applied *-un-lft-identity58.5
  \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\]
5. Applied times-frac58.5
  \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]
6. Applied sqrt-prod58.5
  \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]
7. Taylor expanded around -inf 15.9
  \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)}\]
if -3.9904763547485593e+124 < x < 9.333719392464218e+137
1. Initial program 30.4
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
if 9.333719392464218e+137 < x
1. Initial program 60.6
  \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
2. Taylor expanded around inf 17.2
  \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]
Recombined 3 regimes into one program.
Final simplification26.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.99047635474855928770919369247857947665 \cdot 10^{124}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 9.333719392464218207997872709553383250537 \cdot 10^{137}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))

Error

Try it out

Target

Derivation

`if x < -3.9904763547485593e+124`

`if -3.9904763547485593e+124 < x < 9.333719392464218e+137`

`if 9.333719392464218e+137 < x`

Reproduce

In
Out