Average Error: 11.5 → 0.1
Time: 21.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x + \frac{-1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x + \frac{-1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r280580 = x;
        double r280581 = y;
        double r280582 = 2.0;
        double r280583 = r280581 * r280582;
        double r280584 = z;
        double r280585 = r280583 * r280584;
        double r280586 = r280584 * r280582;
        double r280587 = r280586 * r280584;
        double r280588 = t;
        double r280589 = r280581 * r280588;
        double r280590 = r280587 - r280589;
        double r280591 = r280585 / r280590;
        double r280592 = r280580 - r280591;
        return r280592;
}
↓
double f(double x, double y, double z, double t) {
        double r280593 = x;
        double r280594 = -1.0;
        double r280595 = z;
        double r280596 = y;
        double r280597 = r280595 / r280596;
        double r280598 = 0.5;
        double r280599 = t;
        double r280600 = r280599 / r280595;
        double r280601 = r280598 * r280600;
        double r280602 = r280597 - r280601;
        double r280603 = r280594 / r280602;
        double r280604 = r280593 + r280603;
        return r280604;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.5
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  2. Simplified1.0

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

  4. Applied sub-neg1.0

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

  5. Simplified1.0

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

  7. Applied neg-mul-11.0

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

  8. Applied associate-/l*1.0

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

  9. Taylor expanded around 0 0.1

    \[\leadsto x + \frac{-1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]

  10. Final simplification0.1

    \[\leadsto x + \frac{-1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2019306 +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)))))