Average Error: 11.6 → 1.0
Time: 14.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{1}{\frac{\frac{z}{t}}{y}}}, y, x\right)\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{1}{\frac{\frac{z}{t}}{y}}}, y, x\right)
double f(double x, double y, double z, double t) {
        double r329174 = x;
        double r329175 = y;
        double r329176 = 2.0;
        double r329177 = r329175 * r329176;
        double r329178 = z;
        double r329179 = r329177 * r329178;
        double r329180 = r329178 * r329176;
        double r329181 = r329180 * r329178;
        double r329182 = t;
        double r329183 = r329175 * r329182;
        double r329184 = r329181 - r329183;
        double r329185 = r329179 / r329184;
        double r329186 = r329174 - r329185;
        return r329186;
}
↓
double f(double x, double y, double z, double t) {
        double r329187 = 2.0;
        double r329188 = z;
        double r329189 = r329188 * r329187;
        double r329190 = 1.0;
        double r329191 = t;
        double r329192 = r329188 / r329191;
        double r329193 = y;
        double r329194 = r329192 / r329193;
        double r329195 = r329190 / r329194;
        double r329196 = r329189 - r329195;
        double r329197 = r329187 / r329196;
        double r329198 = -r329197;
        double r329199 = x;
        double r329200 = fma(r329198, r329193, r329199);
        return r329200;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.6
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Simplified3.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{y \cdot t}{z}}, y, x\right)}\]
  3. Using strategy rm

  4. Applied associate-/l*1.0

    \[\leadsto \mathsf{fma}\left(-\frac{2}{z \cdot 2 - \color{blue}{\frac{y}{\frac{z}{t}}}}, y, x\right)\]
  5. Using strategy rm

  6. Applied clear-num1.0

    \[\leadsto \mathsf{fma}\left(-\frac{2}{z \cdot 2 - \color{blue}{\frac{1}{\frac{\frac{z}{t}}{y}}}}, y, x\right)\]

  7. Final simplification1.0

    \[\leadsto \mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{1}{\frac{\frac{z}{t}}{y}}}, y, x\right)\]

Reproduce

herbie shell --seed 2019235 +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)))))