Average Error: 12.0 → 0.1
Time: 19.8s
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 r307566 = x;
        double r307567 = y;
        double r307568 = 2.0;
        double r307569 = r307567 * r307568;
        double r307570 = z;
        double r307571 = r307569 * r307570;
        double r307572 = r307570 * r307568;
        double r307573 = r307572 * r307570;
        double r307574 = t;
        double r307575 = r307567 * r307574;
        double r307576 = r307573 - r307575;
        double r307577 = r307571 / r307576;
        double r307578 = r307566 - r307577;
        return r307578;
}
↓
double f(double x, double y, double z, double t) {
        double r307579 = x;
        double r307580 = 1.0;
        double r307581 = z;
        double r307582 = y;
        double r307583 = r307581 / r307582;
        double r307584 = 0.5;
        double r307585 = t;
        double r307586 = r307585 / r307581;
        double r307587 = r307584 * r307586;
        double r307588 = r307583 - r307587;
        double r307589 = r307580 / r307588;
        double r307590 = r307579 - r307589;
        return r307590;
}

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

Original12.0
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.0

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

  2. Simplified1.1

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

  4. Applied clear-num1.2

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

  5. Taylor expanded around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019347 +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)))))