Average Error: 11.4 → 0.9
Time: 10.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\]
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{y}{2}, \frac{-t}{z}, z\right)}
double f(double x, double y, double z, double t) {
        double r414094 = x;
        double r414095 = y;
        double r414096 = 2.0;
        double r414097 = r414095 * r414096;
        double r414098 = z;
        double r414099 = r414097 * r414098;
        double r414100 = r414098 * r414096;
        double r414101 = r414100 * r414098;
        double r414102 = t;
        double r414103 = r414095 * r414102;
        double r414104 = r414101 - r414103;
        double r414105 = r414099 / r414104;
        double r414106 = r414094 - r414105;
        return r414106;
}
↓
double f(double x, double y, double z, double t) {
        double r414107 = x;
        double r414108 = y;
        double r414109 = 2.0;
        double r414110 = r414108 / r414109;
        double r414111 = t;
        double r414112 = -r414111;
        double r414113 = z;
        double r414114 = r414112 / r414113;
        double r414115 = fma(r414110, r414114, r414113);
        double r414116 = r414108 / r414115;
        double r414117 = r414107 - r414116;
        return r414117;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.4
Target0.1
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.4

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

  2. Simplified0.9

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

  4. Applied *-un-lft-identity0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2020047 +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)))))