Average Error: 11.8 → 0.1
Time: 4.2s
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 r567381 = x;
        double r567382 = y;
        double r567383 = 2.0;
        double r567384 = r567382 * r567383;
        double r567385 = z;
        double r567386 = r567384 * r567385;
        double r567387 = r567385 * r567383;
        double r567388 = r567387 * r567385;
        double r567389 = t;
        double r567390 = r567382 * r567389;
        double r567391 = r567388 - r567390;
        double r567392 = r567386 / r567391;
        double r567393 = r567381 - r567392;
        return r567393;
}
↓
double f(double x, double y, double z, double t) {
        double r567394 = x;
        double r567395 = 1.0;
        double r567396 = 2.0;
        double r567397 = z;
        double r567398 = y;
        double r567399 = r567397 / r567398;
        double r567400 = r567396 * r567399;
        double r567401 = t;
        double r567402 = r567401 / r567397;
        double r567403 = r567400 - r567402;
        double r567404 = r567403 / r567396;
        double r567405 = r567395 / r567404;
        double r567406 = r567394 - r567405;
        return r567406;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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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 
(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)))))