Average Error: 11.7 → 2.7
Time: 4.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}\]
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 \cdot y}{z}}
double f(double x, double y, double z, double t) {
        double r481720 = x;
        double r481721 = y;
        double r481722 = 2.0;
        double r481723 = r481721 * r481722;
        double r481724 = z;
        double r481725 = r481723 * r481724;
        double r481726 = r481724 * r481722;
        double r481727 = r481726 * r481724;
        double r481728 = t;
        double r481729 = r481721 * r481728;
        double r481730 = r481727 - r481729;
        double r481731 = r481725 / r481730;
        double r481732 = r481720 - r481731;
        return r481732;
}
↓
double f(double x, double y, double z, double t) {
        double r481733 = x;
        double r481734 = y;
        double r481735 = 2.0;
        double r481736 = r481734 * r481735;
        double r481737 = z;
        double r481738 = r481737 * r481735;
        double r481739 = t;
        double r481740 = r481739 * r481734;
        double r481741 = r481740 / r481737;
        double r481742 = r481738 - r481741;
        double r481743 = r481736 / r481742;
        double r481744 = r481733 - r481743;
        return r481744;
}

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

Original11.7
Target0.1
Herbie2.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  3. Applied associate-/l*6.7

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

  5. Applied div-sub6.7

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

  6. Simplified2.7

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

  7. Simplified2.7

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

  8. Final simplification2.7

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

Reproduce

herbie shell --seed 2019354 
(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)))))