Average Error: 11.7 → 2.3
Time: 3.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t}{\frac{z}{y}}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - \frac{t}{\frac{z}{y}}}{2}}
double f(double x, double y, double z, double t) {
        double r464172 = x;
        double r464173 = y;
        double r464174 = 2.0;
        double r464175 = r464173 * r464174;
        double r464176 = z;
        double r464177 = r464175 * r464176;
        double r464178 = r464176 * r464174;
        double r464179 = r464178 * r464176;
        double r464180 = t;
        double r464181 = r464173 * r464180;
        double r464182 = r464179 - r464181;
        double r464183 = r464177 / r464182;
        double r464184 = r464172 - r464183;
        return r464184;
}
↓
double f(double x, double y, double z, double t) {
        double r464185 = x;
        double r464186 = y;
        double r464187 = 2.0;
        double r464188 = z;
        double r464189 = r464187 * r464188;
        double r464190 = t;
        double r464191 = r464188 / r464186;
        double r464192 = r464190 / r464191;
        double r464193 = r464189 - r464192;
        double r464194 = r464193 / r464187;
        double r464195 = r464186 / r464194;
        double r464196 = r464185 - r464195;
        return r464196;
}

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
Herbie2.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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*7.0

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

  5. Applied associate-/l*7.0

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

  6. Simplified3.0

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

  8. Applied associate-/l*2.3

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

  9. Final simplification2.3

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

Reproduce

herbie shell --seed 2020056 
(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)))))