Average Error: 11.6 → 2.3
Time: 3.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - t \cdot \frac{y}{z}\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - t \cdot \frac{y}{z}\right)}
double f(double x, double y, double z, double t) {
        double r512241 = x;
        double r512242 = y;
        double r512243 = 2.0;
        double r512244 = r512242 * r512243;
        double r512245 = z;
        double r512246 = r512244 * r512245;
        double r512247 = r512245 * r512243;
        double r512248 = r512247 * r512245;
        double r512249 = t;
        double r512250 = r512242 * r512249;
        double r512251 = r512248 - r512250;
        double r512252 = r512246 / r512251;
        double r512253 = r512241 - r512252;
        return r512253;
}
↓
double f(double x, double y, double z, double t) {
        double r512254 = x;
        double r512255 = y;
        double r512256 = 2.0;
        double r512257 = r512255 * r512256;
        double r512258 = 1.0;
        double r512259 = z;
        double r512260 = r512256 * r512259;
        double r512261 = t;
        double r512262 = r512255 / r512259;
        double r512263 = r512261 * r512262;
        double r512264 = r512260 - r512263;
        double r512265 = r512258 * r512264;
        double r512266 = r512257 / r512265;
        double r512267 = r512254 - r512266;
        return r512267;
}

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*6.7

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

  5. Applied *-un-lft-identity6.7

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

  6. Applied *-un-lft-identity6.7

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

  7. Applied times-frac6.7

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

  8. Simplified6.7

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

  9. Simplified3.0

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \color{blue}{\left(2 \cdot z - \frac{t \cdot y}{z}\right)}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity3.0

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

  12. Applied times-frac2.3

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

  13. Simplified2.3

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

  14. Final simplification2.3

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - t \cdot \frac{y}{z}\right)}\]

Reproduce

herbie shell --seed 2020046 +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)))))