Average Error: 11.6 → 1.3
Time: 20.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}
double f(double x, double y, double z, double t) {
        double r350746 = x;
        double r350747 = y;
        double r350748 = 2.0;
        double r350749 = r350747 * r350748;
        double r350750 = z;
        double r350751 = r350749 * r350750;
        double r350752 = r350750 * r350748;
        double r350753 = r350752 * r350750;
        double r350754 = t;
        double r350755 = r350747 * r350754;
        double r350756 = r350753 - r350755;
        double r350757 = r350751 / r350756;
        double r350758 = r350746 - r350757;
        return r350758;
}
↓
double f(double x, double y, double z, double t) {
        double r350759 = x;
        double r350760 = z;
        double r350761 = y;
        double r350762 = r350761 / r350760;
        double r350763 = r350760 / r350762;
        double r350764 = t;
        double r350765 = 2.0;
        double r350766 = r350764 / r350765;
        double r350767 = r350763 - r350766;
        double r350768 = r350760 / r350767;
        double r350769 = r350759 - r350768;
        return r350769;
}

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. Final simplification1.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))