Average Error: 11.6 → 0.1
Time: 17.8s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]
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{0.5}{\frac{z}{t}}}
double f(double x, double y, double z, double t) {
        double r23898112 = x;
        double r23898113 = y;
        double r23898114 = 2.0;
        double r23898115 = r23898113 * r23898114;
        double r23898116 = z;
        double r23898117 = r23898115 * r23898116;
        double r23898118 = r23898116 * r23898114;
        double r23898119 = r23898118 * r23898116;
        double r23898120 = t;
        double r23898121 = r23898113 * r23898120;
        double r23898122 = r23898119 - r23898121;
        double r23898123 = r23898117 / r23898122;
        double r23898124 = r23898112 - r23898123;
        return r23898124;
}
↓
double f(double x, double y, double z, double t) {
        double r23898125 = x;
        double r23898126 = 1.0;
        double r23898127 = z;
        double r23898128 = y;
        double r23898129 = r23898127 / r23898128;
        double r23898130 = 0.5;
        double r23898131 = t;
        double r23898132 = r23898127 / r23898131;
        double r23898133 = r23898130 / r23898132;
        double r23898134 = r23898129 - r23898133;
        double r23898135 = r23898126 / r23898134;
        double r23898136 = r23898125 - r23898135;
        return r23898136;
}

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. Simplified1.0

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

  4. Applied clear-num1.0

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

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]
  6. Using strategy rm

  7. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}{1}}}\]

  8. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}}\]

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]

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)))))