Average Error: 11.8 → 0.1
Time: 5.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{2 \cdot \frac{z}{y} - \frac{t}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{2 \cdot \frac{z}{y} - \frac{t}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r461212 = x;
        double r461213 = y;
        double r461214 = 2.0;
        double r461215 = r461213 * r461214;
        double r461216 = z;
        double r461217 = r461215 * r461216;
        double r461218 = r461216 * r461214;
        double r461219 = r461218 * r461216;
        double r461220 = t;
        double r461221 = r461213 * r461220;
        double r461222 = r461219 - r461221;
        double r461223 = r461217 / r461222;
        double r461224 = r461212 - r461223;
        return r461224;
}
↓
double f(double x, double y, double z, double t) {
        double r461225 = x;
        double r461226 = 1.0;
        double r461227 = 2.0;
        double r461228 = z;
        double r461229 = y;
        double r461230 = r461228 / r461229;
        double r461231 = r461227 * r461230;
        double r461232 = t;
        double r461233 = r461232 / r461228;
        double r461234 = r461231 - r461233;
        double r461235 = r461234 / r461227;
        double r461236 = r461226 / r461235;
        double r461237 = r461225 - r461236;
        return r461237;
}

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

Original11.8
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.8

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

  3. Applied associate-/l*6.8

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

  5. Applied *-un-lft-identity6.8

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

  6. Applied *-un-lft-identity6.8

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

  7. Applied times-frac6.8

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

  8. Simplified6.8

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

  9. Simplified2.9

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

  11. Applied clear-num3.0

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

  12. Simplified2.9

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

  13. Taylor expanded around 0 0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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)))))