Average Error: 11.8 → 1.1
Time: 20.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}
double f(double x, double y, double z, double t) {
        double r368932 = x;
        double r368933 = y;
        double r368934 = 2.0;
        double r368935 = r368933 * r368934;
        double r368936 = z;
        double r368937 = r368935 * r368936;
        double r368938 = r368936 * r368934;
        double r368939 = r368938 * r368936;
        double r368940 = t;
        double r368941 = r368933 * r368940;
        double r368942 = r368939 - r368941;
        double r368943 = r368937 / r368942;
        double r368944 = r368932 - r368943;
        return r368944;
}
↓
double f(double x, double y, double z, double t) {
        double r368945 = x;
        double r368946 = y;
        double r368947 = t;
        double r368948 = z;
        double r368949 = r368947 / r368948;
        double r368950 = 2.0;
        double r368951 = r368946 / r368950;
        double r368952 = -r368951;
        double r368953 = fma(r368949, r368952, r368948);
        double r368954 = r368946 / r368953;
        double r368955 = r368945 - r368954;
        return r368955;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.8
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.8

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

  2. Simplified1.1

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

  3. Final simplification1.1

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

Reproduce

herbie shell --seed 2019303 +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)))))