Average Error: 12.1 → 1.0
Time: 13.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{y}{\frac{-y}{\frac{z}{t}} + 2 \cdot z}, -2, x\right)\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\mathsf{fma}\left(\frac{y}{\frac{-y}{\frac{z}{t}} + 2 \cdot z}, -2, x\right)
double f(double x, double y, double z, double t) {
        double r365878 = x;
        double r365879 = y;
        double r365880 = 2.0;
        double r365881 = r365879 * r365880;
        double r365882 = z;
        double r365883 = r365881 * r365882;
        double r365884 = r365882 * r365880;
        double r365885 = r365884 * r365882;
        double r365886 = t;
        double r365887 = r365879 * r365886;
        double r365888 = r365885 - r365887;
        double r365889 = r365883 / r365888;
        double r365890 = r365878 - r365889;
        return r365890;
}
↓
double f(double x, double y, double z, double t) {
        double r365891 = y;
        double r365892 = -r365891;
        double r365893 = z;
        double r365894 = t;
        double r365895 = r365893 / r365894;
        double r365896 = r365892 / r365895;
        double r365897 = 2.0;
        double r365898 = r365897 * r365893;
        double r365899 = r365896 + r365898;
        double r365900 = r365891 / r365899;
        double r365901 = -r365897;
        double r365902 = x;
        double r365903 = fma(r365900, r365901, r365902);
        return r365903;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original12.1
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  2. Simplified2.5

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

  3. Taylor expanded around 0 2.8

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

  4. Simplified1.1

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

  6. Applied fma-udef1.1

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

  7. Simplified1.0

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

  8. Final simplification1.0

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

Reproduce

herbie shell --seed 2019196 +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)))))