Average Error: 11.6 → 1.3
Time: 15.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{z \cdot \frac{1}{\frac{y}{z}} - \frac{t}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{z \cdot \frac{1}{\frac{y}{z}} - \frac{t}{2}}
double f(double x, double y, double z, double t) {
        double r574701 = x;
        double r574702 = y;
        double r574703 = 2.0;
        double r574704 = r574702 * r574703;
        double r574705 = z;
        double r574706 = r574704 * r574705;
        double r574707 = r574705 * r574703;
        double r574708 = r574707 * r574705;
        double r574709 = t;
        double r574710 = r574702 * r574709;
        double r574711 = r574708 - r574710;
        double r574712 = r574706 / r574711;
        double r574713 = r574701 - r574712;
        return r574713;
}
↓
double f(double x, double y, double z, double t) {
        double r574714 = x;
        double r574715 = z;
        double r574716 = 1.0;
        double r574717 = y;
        double r574718 = r574717 / r574715;
        double r574719 = r574716 / r574718;
        double r574720 = r574715 * r574719;
        double r574721 = t;
        double r574722 = 2.0;
        double r574723 = r574721 / r574722;
        double r574724 = r574720 - r574723;
        double r574725 = r574715 / r574724;
        double r574726 = r574714 - r574725;
        return r574726;
}

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

Original11.6
Target0.1
Herbie1.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified3.5

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

  4. Applied associate-/l*1.3

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

  6. Applied div-inv1.3

    \[\leadsto x - \frac{z}{\color{blue}{z \cdot \frac{1}{\frac{y}{z}}} - \frac{t}{2}}\]

  7. Final simplification1.3

    \[\leadsto x - \frac{z}{z \cdot \frac{1}{\frac{y}{z}} - \frac{t}{2}}\]

Reproduce

herbie shell --seed 2020046 +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)))))