Average Error: 11.9 → 0.1
Time: 19.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}
double f(double x, double y, double z, double t) {
        double r19612089 = x;
        double r19612090 = y;
        double r19612091 = 2.0;
        double r19612092 = r19612090 * r19612091;
        double r19612093 = z;
        double r19612094 = r19612092 * r19612093;
        double r19612095 = r19612093 * r19612091;
        double r19612096 = r19612095 * r19612093;
        double r19612097 = t;
        double r19612098 = r19612090 * r19612097;
        double r19612099 = r19612096 - r19612098;
        double r19612100 = r19612094 / r19612099;
        double r19612101 = r19612089 - r19612100;
        return r19612101;
}
↓
double f(double x, double y, double z, double t) {
        double r19612102 = x;
        double r19612103 = 1.0;
        double r19612104 = z;
        double r19612105 = y;
        double r19612106 = r19612104 / r19612105;
        double r19612107 = t;
        double r19612108 = r19612107 / r19612104;
        double r19612109 = 0.5;
        double r19612110 = r19612108 * r19612109;
        double r19612111 = r19612106 - r19612110;
        double r19612112 = r19612103 / r19612111;
        double r19612113 = r19612102 - r19612112;
        return r19612113;
}

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

Original11.9
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  2. Simplified0.9

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

  4. Applied clear-num1.0

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

  5. Taylor expanded around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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