Average Error: 11.6 → 0.1
Time: 16.2s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}
double f(double x, double y, double z, double t) {
        double r24817761 = x;
        double r24817762 = y;
        double r24817763 = 2.0;
        double r24817764 = r24817762 * r24817763;
        double r24817765 = z;
        double r24817766 = r24817764 * r24817765;
        double r24817767 = r24817765 * r24817763;
        double r24817768 = r24817767 * r24817765;
        double r24817769 = t;
        double r24817770 = r24817762 * r24817769;
        double r24817771 = r24817768 - r24817770;
        double r24817772 = r24817766 / r24817771;
        double r24817773 = r24817761 - r24817772;
        return r24817773;
}
↓
double f(double x, double y, double z, double t) {
        double r24817774 = x;
        double r24817775 = 1.0;
        double r24817776 = z;
        double r24817777 = y;
        double r24817778 = r24817776 / r24817777;
        double r24817779 = t;
        double r24817780 = r24817779 / r24817776;
        double r24817781 = 0.5;
        double r24817782 = r24817780 * r24817781;
        double r24817783 = r24817778 - r24817782;
        double r24817784 = r24817775 / r24817783;
        double r24817785 = r24817774 - r24817784;
        return r24817785;
}

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

Original11.6
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.6

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.0

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{\frac{t}{z} \cdot y}{2.0}}}\]
  3. Using strategy rm

  4. Applied clear-num1.0

    \[\leadsto x - \color{blue}{\frac{1}{\frac{z - \frac{\frac{t}{z} \cdot y}{2.0}}{y}}}\]

  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

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))