Average Error: 11.7 → 2.2
Time: 3.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r519657 = x;
        double r519658 = y;
        double r519659 = 2.0;
        double r519660 = r519658 * r519659;
        double r519661 = z;
        double r519662 = r519660 * r519661;
        double r519663 = r519661 * r519659;
        double r519664 = r519663 * r519661;
        double r519665 = t;
        double r519666 = r519658 * r519665;
        double r519667 = r519664 - r519666;
        double r519668 = r519662 / r519667;
        double r519669 = r519657 - r519668;
        return r519669;
}
↓
double f(double x, double y, double z, double t) {
        double r519670 = x;
        double r519671 = y;
        double r519672 = 2.0;
        double r519673 = z;
        double r519674 = r519672 * r519673;
        double r519675 = t;
        double r519676 = r519671 / r519673;
        double r519677 = r519675 * r519676;
        double r519678 = r519674 - r519677;
        double r519679 = r519678 / r519672;
        double r519680 = r519671 / r519679;
        double r519681 = r519670 - r519680;
        return r519681;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

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Target

Original11.7
Target0.1
Herbie2.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  3. Applied associate-/l*6.6

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

  5. Applied associate-/l*6.6

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

  6. Simplified2.7

    \[\leadsto x - \frac{y}{\color{blue}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity2.7

    \[\leadsto x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{\color{blue}{1 \cdot z}}}{2}}\]

  9. Applied times-frac2.2

    \[\leadsto x - \frac{y}{\frac{2 \cdot z - \color{blue}{\frac{t}{1} \cdot \frac{y}{z}}}{2}}\]

  10. Simplified2.2

    \[\leadsto x - \frac{y}{\frac{2 \cdot z - \color{blue}{t} \cdot \frac{y}{z}}{2}}\]

  11. Final simplification2.2

    \[\leadsto x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}\]

Reproduce

herbie shell --seed 2020025 
(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)))))