Average Error: 11.6 → 0.1
Time: 13.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{z}{y}, 2, -\frac{t}{z}\right)}, -2, x\right)\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{z}{y}, 2, -\frac{t}{z}\right)}, -2, x\right)
double f(double x, double y, double z, double t) {
        double r342164 = x;
        double r342165 = y;
        double r342166 = 2.0;
        double r342167 = r342165 * r342166;
        double r342168 = z;
        double r342169 = r342167 * r342168;
        double r342170 = r342168 * r342166;
        double r342171 = r342170 * r342168;
        double r342172 = t;
        double r342173 = r342165 * r342172;
        double r342174 = r342171 - r342173;
        double r342175 = r342169 / r342174;
        double r342176 = r342164 - r342175;
        return r342176;
}
↓
double f(double x, double y, double z, double t) {
        double r342177 = 1.0;
        double r342178 = z;
        double r342179 = y;
        double r342180 = r342178 / r342179;
        double r342181 = 2.0;
        double r342182 = t;
        double r342183 = r342182 / r342178;
        double r342184 = -r342183;
        double r342185 = fma(r342180, r342181, r342184);
        double r342186 = r342177 / r342185;
        double r342187 = -r342181;
        double r342188 = x;
        double r342189 = fma(r342186, r342187, r342188);
        return r342189;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.6
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified2.2

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

  4. Applied clear-num2.2

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

  5. Simplified2.2

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

  6. Taylor expanded around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +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)))))