Average Error: 11.2 → 0.1
Time: 15.2s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]
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} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}
double f(double x, double y, double z, double t) {
        double r25984243 = x;
        double r25984244 = y;
        double r25984245 = 2.0;
        double r25984246 = r25984244 * r25984245;
        double r25984247 = z;
        double r25984248 = r25984246 * r25984247;
        double r25984249 = r25984247 * r25984245;
        double r25984250 = r25984249 * r25984247;
        double r25984251 = t;
        double r25984252 = r25984244 * r25984251;
        double r25984253 = r25984250 - r25984252;
        double r25984254 = r25984248 / r25984253;
        double r25984255 = r25984243 - r25984254;
        return r25984255;
}
↓
double f(double x, double y, double z, double t) {
        double r25984256 = x;
        double r25984257 = 1.0;
        double r25984258 = z;
        double r25984259 = y;
        double r25984260 = r25984258 / r25984259;
        double r25984261 = 0.5;
        double r25984262 = t;
        double r25984263 = r25984261 * r25984262;
        double r25984264 = r25984257 / r25984258;
        double r25984265 = r25984263 * r25984264;
        double r25984266 = r25984260 - r25984265;
        double r25984267 = r25984257 / r25984266;
        double r25984268 = r25984256 - r25984267;
        return r25984268;
}

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

Original11.2
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.2

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.0

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

  4. Applied clear-num1.0

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

  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 div-inv0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]

Reproduce

herbie shell --seed 2019162 
(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)))))