Average Error: 11.6 → 1.3
Time: 19.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}
double f(double x, double y, double z, double t) {
        double r320063 = x;
        double r320064 = y;
        double r320065 = 2.0;
        double r320066 = r320064 * r320065;
        double r320067 = z;
        double r320068 = r320066 * r320067;
        double r320069 = r320067 * r320065;
        double r320070 = r320069 * r320067;
        double r320071 = t;
        double r320072 = r320064 * r320071;
        double r320073 = r320070 - r320072;
        double r320074 = r320068 / r320073;
        double r320075 = r320063 - r320074;
        return r320075;
}
↓
double f(double x, double y, double z, double t) {
        double r320076 = x;
        double r320077 = z;
        double r320078 = y;
        double r320079 = r320078 / r320077;
        double r320080 = r320077 / r320079;
        double r320081 = t;
        double r320082 = 2.0;
        double r320083 = r320081 / r320082;
        double r320084 = r320080 - r320083;
        double r320085 = r320077 / r320084;
        double r320086 = r320076 - r320085;
        return r320086;
}

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
Herbie1.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified3.5

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

  4. Applied associate-/l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))