Average Error: 11.6 → 0.1
Time: 34.1s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}
double f(double x, double y, double z, double t) {
        double r19778122 = x;
        double r19778123 = y;
        double r19778124 = 2.0;
        double r19778125 = r19778123 * r19778124;
        double r19778126 = z;
        double r19778127 = r19778125 * r19778126;
        double r19778128 = r19778126 * r19778124;
        double r19778129 = r19778128 * r19778126;
        double r19778130 = t;
        double r19778131 = r19778123 * r19778130;
        double r19778132 = r19778129 - r19778131;
        double r19778133 = r19778127 / r19778132;
        double r19778134 = r19778122 - r19778133;
        return r19778134;
}
↓
double f(double x, double y, double z, double t) {
        double r19778135 = x;
        double r19778136 = 1.0;
        double r19778137 = z;
        double r19778138 = y;
        double r19778139 = r19778137 / r19778138;
        double r19778140 = t;
        double r19778141 = 2.0;
        double r19778142 = r19778140 / r19778141;
        double r19778143 = r19778142 / r19778137;
        double r19778144 = r19778139 - r19778143;
        double r19778145 = r19778136 / r19778144;
        double r19778146 = r19778135 - r19778145;
        return r19778146;
}

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

Original11.6
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.6

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified2.1

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{\frac{t}{2.0}}{\frac{z}{y}}}}\]
  3. Using strategy rm

  4. Applied associate-/r/1.0

    \[\leadsto x - \frac{y}{z - \color{blue}{\frac{\frac{t}{2.0}}{z} \cdot y}}\]
  5. Using strategy rm

  6. Applied clear-num1.0

    \[\leadsto x - \color{blue}{\frac{1}{\frac{z - \frac{\frac{t}{2.0}}{z} \cdot y}{y}}}\]

  7. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}}\]

  8. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))