Average Error: 11.8 → 1.1
Time: 21.8s
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 r324999 = x;
        double r325000 = y;
        double r325001 = 2.0;
        double r325002 = r325000 * r325001;
        double r325003 = z;
        double r325004 = r325002 * r325003;
        double r325005 = r325003 * r325001;
        double r325006 = r325005 * r325003;
        double r325007 = t;
        double r325008 = r325000 * r325007;
        double r325009 = r325006 - r325008;
        double r325010 = r325004 / r325009;
        double r325011 = r324999 - r325010;
        return r325011;
}
↓
double f(double x, double y, double z, double t) {
        double r325012 = x;
        double r325013 = y;
        double r325014 = t;
        double r325015 = z;
        double r325016 = r325014 / r325015;
        double r325017 = 2.0;
        double r325018 = r325013 / r325017;
        double r325019 = -r325018;
        double r325020 = fma(r325016, r325019, r325015);
        double r325021 = r325013 / r325020;
        double r325022 = r325012 - r325021;
        return r325022;
}

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)))))