Average Error: 11.5 → 2.6
Time: 4.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r541097 = x;
        double r541098 = y;
        double r541099 = 2.0;
        double r541100 = r541098 * r541099;
        double r541101 = z;
        double r541102 = r541100 * r541101;
        double r541103 = r541101 * r541099;
        double r541104 = r541103 * r541101;
        double r541105 = t;
        double r541106 = r541098 * r541105;
        double r541107 = r541104 - r541106;
        double r541108 = r541102 / r541107;
        double r541109 = r541097 - r541108;
        return r541109;
}
↓
double f(double x, double y, double z, double t) {
        double r541110 = x;
        double r541111 = y;
        double r541112 = 2.0;
        double r541113 = z;
        double r541114 = r541112 * r541113;
        double r541115 = t;
        double r541116 = r541115 * r541111;
        double r541117 = r541116 / r541113;
        double r541118 = r541114 - r541117;
        double r541119 = r541118 / r541112;
        double r541120 = r541111 / r541119;
        double r541121 = r541110 - r541120;
        return r541121;
}

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
Herbie2.6
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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 associate-/l*6.6

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

  6. Simplified2.6

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

  7. Final simplification2.6

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

Reproduce

herbie shell --seed 2020003 
(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)))))