Average Error: 11.6 → 1.0
Time: 22.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{y}{\frac{z}{t}}}, y, x\right)\]
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{y}{\frac{z}{t}}}, y, x\right)
double f(double x, double y, double z, double t) {
        double r377239 = x;
        double r377240 = y;
        double r377241 = 2.0;
        double r377242 = r377240 * r377241;
        double r377243 = z;
        double r377244 = r377242 * r377243;
        double r377245 = r377243 * r377241;
        double r377246 = r377245 * r377243;
        double r377247 = t;
        double r377248 = r377240 * r377247;
        double r377249 = r377246 - r377248;
        double r377250 = r377244 / r377249;
        double r377251 = r377239 - r377250;
        return r377251;
}
↓
double f(double x, double y, double z, double t) {
        double r377252 = 2.0;
        double r377253 = z;
        double r377254 = r377253 * r377252;
        double r377255 = y;
        double r377256 = t;
        double r377257 = r377253 / r377256;
        double r377258 = r377255 / r377257;
        double r377259 = r377254 - r377258;
        double r377260 = r377252 / r377259;
        double r377261 = -r377260;
        double r377262 = x;
        double r377263 = fma(r377261, r377255, r377262);
        return r377263;
}

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. Simplified2.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{2}{z \cdot 2 - \frac{y \cdot t}{z}}, y, x\right)}\]
  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. Final simplification1.0

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

Reproduce

herbie shell --seed 2019199 +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)))))