Average Error: 12.1 → 1.5
Time: 3.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}
double f(double x, double y, double z, double t) {
        double r478993 = x;
        double r478994 = y;
        double r478995 = 2.0;
        double r478996 = r478994 * r478995;
        double r478997 = z;
        double r478998 = r478996 * r478997;
        double r478999 = r478997 * r478995;
        double r479000 = r478999 * r478997;
        double r479001 = t;
        double r479002 = r478994 * r479001;
        double r479003 = r479000 - r479002;
        double r479004 = r478998 / r479003;
        double r479005 = r478993 - r479004;
        return r479005;
}
↓
double f(double x, double y, double z, double t) {
        double r479006 = x;
        double r479007 = y;
        double r479008 = 2.0;
        double r479009 = r479007 * r479008;
        double r479010 = z;
        double r479011 = r479010 * r479008;
        double r479012 = t;
        double r479013 = cbrt(r479010);
        double r479014 = r479013 * r479013;
        double r479015 = r479012 / r479014;
        double r479016 = r479007 / r479013;
        double r479017 = r479015 * r479016;
        double r479018 = r479011 - r479017;
        double r479019 = r479009 / r479018;
        double r479020 = r479006 - r479019;
        return r479020;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.1
Target0.1
Herbie1.5
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

    \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
  2. Using strategy rm

  3. Applied associate-/l*6.8

    \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\]
  4. Using strategy rm

  5. Applied div-sub6.8

    \[\leadsto x - \frac{y \cdot 2}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z}{z} - \frac{y \cdot t}{z}}}\]

  6. Simplified3.1

    \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2} - \frac{y \cdot t}{z}}\]

  7. Simplified3.1

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t \cdot y}{z}}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt3.2

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\]

  10. Applied times-frac1.5

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}}\]

  11. Final simplification1.5

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))