Average Error: 11.8 → 0.1
Time: 4.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{1 \cdot \frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{1 \cdot \frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r453112 = x;
        double r453113 = y;
        double r453114 = 2.0;
        double r453115 = r453113 * r453114;
        double r453116 = z;
        double r453117 = r453115 * r453116;
        double r453118 = r453116 * r453114;
        double r453119 = r453118 * r453116;
        double r453120 = t;
        double r453121 = r453113 * r453120;
        double r453122 = r453119 - r453121;
        double r453123 = r453117 / r453122;
        double r453124 = r453112 - r453123;
        return r453124;
}
↓
double f(double x, double y, double z, double t) {
        double r453125 = x;
        double r453126 = 1.0;
        double r453127 = 1.0;
        double r453128 = z;
        double r453129 = y;
        double r453130 = r453128 / r453129;
        double r453131 = r453127 * r453130;
        double r453132 = 0.5;
        double r453133 = t;
        double r453134 = r453133 / r453128;
        double r453135 = r453132 * r453134;
        double r453136 = r453131 - r453135;
        double r453137 = r453126 / r453136;
        double r453138 = r453125 - r453137;
        return r453138;
}

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

Original11.8
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 associate-/l*6.7

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

  6. Simplified2.9

    \[\leadsto x - \frac{y}{\color{blue}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}}\]
  7. Using strategy rm

  8. Applied clear-num2.9

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}{y}}}\]

  9. Taylor expanded around 0 0.1

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

  10. Final simplification0.1

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

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)))))