Average Error: 11.7 → 1.2
Time: 3.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t}{z} \cdot y}{2}}\]
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}{z} \cdot y}{2}}
double f(double x, double y, double z, double t) {
        double r451209 = x;
        double r451210 = y;
        double r451211 = 2.0;
        double r451212 = r451210 * r451211;
        double r451213 = z;
        double r451214 = r451212 * r451213;
        double r451215 = r451213 * r451211;
        double r451216 = r451215 * r451213;
        double r451217 = t;
        double r451218 = r451210 * r451217;
        double r451219 = r451216 - r451218;
        double r451220 = r451214 / r451219;
        double r451221 = r451209 - r451220;
        return r451221;
}
↓
double f(double x, double y, double z, double t) {
        double r451222 = x;
        double r451223 = y;
        double r451224 = 2.0;
        double r451225 = z;
        double r451226 = r451224 * r451225;
        double r451227 = t;
        double r451228 = r451227 / r451225;
        double r451229 = r451228 * r451223;
        double r451230 = r451226 - r451229;
        double r451231 = r451230 / r451224;
        double r451232 = r451223 / r451231;
        double r451233 = r451222 - r451232;
        return r451233;
}

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

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*6.9

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

  5. Applied associate-/l*6.9

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

  6. Simplified3.1

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

  8. Applied associate-/l*2.4

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

  10. Applied associate-/r/1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2020062 
(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)))))