Average Error: 11.3 → 0.1
Time: 12.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r366501 = x;
        double r366502 = y;
        double r366503 = 2.0;
        double r366504 = r366502 * r366503;
        double r366505 = z;
        double r366506 = r366504 * r366505;
        double r366507 = r366505 * r366503;
        double r366508 = r366507 * r366505;
        double r366509 = t;
        double r366510 = r366502 * r366509;
        double r366511 = r366508 - r366510;
        double r366512 = r366506 / r366511;
        double r366513 = r366501 - r366512;
        return r366513;
}
↓
double f(double x, double y, double z, double t) {
        double r366514 = x;
        double r366515 = 1.0;
        double r366516 = z;
        double r366517 = y;
        double r366518 = r366516 / r366517;
        double r366519 = 0.5;
        double r366520 = t;
        double r366521 = r366520 / r366516;
        double r366522 = r366519 * r366521;
        double r366523 = r366518 - r366522;
        double r366524 = r366515 / r366523;
        double r366525 = r366514 - r366524;
        return r366525;
}

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

Original11.3
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified2.6

    \[\leadsto \color{blue}{x - \frac{y}{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}}\]
  3. Using strategy rm

  4. Applied associate-/l*0.9

    \[\leadsto x - \frac{y}{\frac{z}{1} - \frac{\color{blue}{\frac{y}{\frac{z}{t}}}}{2}}\]
  5. Using strategy rm

  6. Applied clear-num0.9

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

  7. Simplified0.9

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

  8. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2019208 
(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)))))