Average Error: 11.4 → 0.9
Time: 7.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 526596793.112873614:\\ \;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;z \le 526596793.112873614:\\
\;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r463063 = x;
        double r463064 = y;
        double r463065 = 2.0;
        double r463066 = r463064 * r463065;
        double r463067 = z;
        double r463068 = r463066 * r463067;
        double r463069 = r463067 * r463065;
        double r463070 = r463069 * r463067;
        double r463071 = t;
        double r463072 = r463064 * r463071;
        double r463073 = r463070 - r463072;
        double r463074 = r463068 / r463073;
        double r463075 = r463063 - r463074;
        return r463075;
}
↓
double f(double x, double y, double z, double t) {
        double r463076 = z;
        double r463077 = 526596793.1128736;
        bool r463078 = r463076 <= r463077;
        double r463079 = x;
        double r463080 = y;
        double r463081 = r463080 / r463076;
        double r463082 = r463076 / r463081;
        double r463083 = t;
        double r463084 = 2.0;
        double r463085 = r463083 / r463084;
        double r463086 = r463082 - r463085;
        double r463087 = r463076 / r463086;
        double r463088 = r463079 - r463087;
        double r463089 = r463076 * r463084;
        double r463090 = r463081 * r463083;
        double r463091 = r463089 - r463090;
        double r463092 = r463076 / r463091;
        double r463093 = r463081 * r463084;
        double r463094 = r463092 * r463093;
        double r463095 = r463079 - r463094;
        double r463096 = r463078 ? r463088 : r463095;
        return r463096;
}

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

Original11.4
Target0.1
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 526596793.1128736

    1. Initial program 9.3

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

    2. Simplified2.8

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

    4. Applied associate-/l*1.0

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

    if 526596793.1128736 < z

    1. Initial program 18.0

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

    2. Simplified6.5

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

    4. Applied associate-/l*3.1

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

    6. Applied frac-sub3.8

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

    7. Applied associate-/r/0.7

      \[\leadsto x - \color{blue}{\frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 526596793.112873614:\\ \;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))))