Average Error: 11.5 → 2.7
Time: 3.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}
double f(double x, double y, double z, double t) {
        double r566100 = x;
        double r566101 = y;
        double r566102 = 2.0;
        double r566103 = r566101 * r566102;
        double r566104 = z;
        double r566105 = r566103 * r566104;
        double r566106 = r566104 * r566102;
        double r566107 = r566106 * r566104;
        double r566108 = t;
        double r566109 = r566101 * r566108;
        double r566110 = r566107 - r566109;
        double r566111 = r566105 / r566110;
        double r566112 = r566100 - r566111;
        return r566112;
}
↓
double f(double x, double y, double z, double t) {
        double r566113 = x;
        double r566114 = y;
        double r566115 = 2.0;
        double r566116 = r566114 * r566115;
        double r566117 = z;
        double r566118 = r566117 * r566115;
        double r566119 = t;
        double r566120 = r566119 * r566114;
        double r566121 = r566120 / r566117;
        double r566122 = r566118 - r566121;
        double r566123 = r566116 / r566122;
        double r566124 = r566113 - r566123;
        return r566124;
}

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

Original11.5
Target0.1
Herbie2.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  3. Applied associate-/l*6.6

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

  5. Applied div-sub6.6

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

  6. Simplified2.7

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

  7. Simplified2.7

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

  8. Final simplification2.7

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

Reproduce

herbie shell --seed 2020027 
(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)))))