Average Error: 11.3 → 1.0
Time: 13.8s
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 r393806 = x;
        double r393807 = y;
        double r393808 = 2.0;
        double r393809 = r393807 * r393808;
        double r393810 = z;
        double r393811 = r393809 * r393810;
        double r393812 = r393810 * r393808;
        double r393813 = r393812 * r393810;
        double r393814 = t;
        double r393815 = r393807 * r393814;
        double r393816 = r393813 - r393815;
        double r393817 = r393811 / r393816;
        double r393818 = r393806 - r393817;
        return r393818;
}
↓
double f(double x, double y, double z, double t) {
        double r393819 = 2.0;
        double r393820 = z;
        double r393821 = r393820 * r393819;
        double r393822 = 1.0;
        double r393823 = t;
        double r393824 = r393820 / r393823;
        double r393825 = y;
        double r393826 = r393824 / r393825;
        double r393827 = r393822 / r393826;
        double r393828 = r393821 - r393827;
        double r393829 = r393819 / r393828;
        double r393830 = -r393829;
        double r393831 = x;
        double r393832 = fma(r393830, r393825, r393831);
        return r393832;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.3
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  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*1.1

    \[\leadsto \mathsf{fma}\left(-\frac{2}{z \cdot 2 - \color{blue}{\frac{y}{\frac{z}{t}}}}, y, x\right)\]
  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 2019198 +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)))))