Average Error: 11.2 → 0.1
Time: 13.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\frac{z}{y}, 2, -\frac{t}{z}\right)}, -2, x\right)\]
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 r386216 = x;
        double r386217 = y;
        double r386218 = 2.0;
        double r386219 = r386217 * r386218;
        double r386220 = z;
        double r386221 = r386219 * r386220;
        double r386222 = r386220 * r386218;
        double r386223 = r386222 * r386220;
        double r386224 = t;
        double r386225 = r386217 * r386224;
        double r386226 = r386223 - r386225;
        double r386227 = r386221 / r386226;
        double r386228 = r386216 - r386227;
        return r386228;
}
↓
double f(double x, double y, double z, double t) {
        double r386229 = 1.0;
        double r386230 = z;
        double r386231 = y;
        double r386232 = r386230 / r386231;
        double r386233 = 2.0;
        double r386234 = t;
        double r386235 = r386234 / r386230;
        double r386236 = -r386235;
        double r386237 = fma(r386232, r386233, r386236);
        double r386238 = r386229 / r386237;
        double r386239 = -r386233;
        double r386240 = x;
        double r386241 = fma(r386238, r386239, r386240);
        return r386241;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 11.2

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

  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. Simplified1.0

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

  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 2019194 +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)))))