Average Error: 11.7 → 3.0
Time: 4.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \frac{t \cdot y}{z}\right)}\]
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 - \frac{t \cdot y}{z}\right)}
double f(double x, double y, double z, double t) {
        double r440104 = x;
        double r440105 = y;
        double r440106 = 2.0;
        double r440107 = r440105 * r440106;
        double r440108 = z;
        double r440109 = r440107 * r440108;
        double r440110 = r440108 * r440106;
        double r440111 = r440110 * r440108;
        double r440112 = t;
        double r440113 = r440105 * r440112;
        double r440114 = r440111 - r440113;
        double r440115 = r440109 / r440114;
        double r440116 = r440104 - r440115;
        return r440116;
}
↓
double f(double x, double y, double z, double t) {
        double r440117 = x;
        double r440118 = y;
        double r440119 = 2.0;
        double r440120 = r440118 * r440119;
        double r440121 = 1.0;
        double r440122 = z;
        double r440123 = r440119 * r440122;
        double r440124 = t;
        double r440125 = r440124 * r440118;
        double r440126 = r440125 / r440122;
        double r440127 = r440123 - r440126;
        double r440128 = r440121 * r440127;
        double r440129 = r440120 / r440128;
        double r440130 = r440117 - r440129;
        return r440130;
}

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
Herbie3.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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 *-un-lft-identity7.0

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

  6. Applied *-un-lft-identity7.0

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

  7. Applied times-frac7.0

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

  8. Simplified7.0

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

  9. Simplified3.0

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

  10. Final simplification3.0

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

Reproduce

herbie shell --seed 2020056 +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)))))