Average Error: 11.6 → 2.3
Time: 4.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - t \cdot \frac{y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - t \cdot \frac{y}{z}}
double f(double x, double y, double z, double t) {
        double r662259 = x;
        double r662260 = y;
        double r662261 = 2.0;
        double r662262 = r662260 * r662261;
        double r662263 = z;
        double r662264 = r662262 * r662263;
        double r662265 = r662263 * r662261;
        double r662266 = r662265 * r662263;
        double r662267 = t;
        double r662268 = r662260 * r662267;
        double r662269 = r662266 - r662268;
        double r662270 = r662264 / r662269;
        double r662271 = r662259 - r662270;
        return r662271;
}
↓
double f(double x, double y, double z, double t) {
        double r662272 = x;
        double r662273 = y;
        double r662274 = 2.0;
        double r662275 = r662273 * r662274;
        double r662276 = z;
        double r662277 = r662274 * r662276;
        double r662278 = t;
        double r662279 = r662273 / r662276;
        double r662280 = r662278 * r662279;
        double r662281 = r662277 - r662280;
        double r662282 = r662275 / r662281;
        double r662283 = r662272 - r662282;
        return r662283;
}

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

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. Using strategy rm

  3. Applied associate-/l*6.4

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

  4. Taylor expanded around 0 2.7

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

  6. Applied *-un-lft-identity2.7

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

  7. Applied times-frac2.3

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

  8. Simplified2.3

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

  9. Final simplification2.3

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

Reproduce

herbie shell --seed 2020100 +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)))))