Average Error: 11.5 → 1.1
Time: 17.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{z - \frac{t}{z} \cdot \frac{y}{2}} \cdot y\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{z - \frac{t}{z} \cdot \frac{y}{2}} \cdot y
double f(double x, double y, double z, double t) {
        double r26906706 = x;
        double r26906707 = y;
        double r26906708 = 2.0;
        double r26906709 = r26906707 * r26906708;
        double r26906710 = z;
        double r26906711 = r26906709 * r26906710;
        double r26906712 = r26906710 * r26906708;
        double r26906713 = r26906712 * r26906710;
        double r26906714 = t;
        double r26906715 = r26906707 * r26906714;
        double r26906716 = r26906713 - r26906715;
        double r26906717 = r26906711 / r26906716;
        double r26906718 = r26906706 - r26906717;
        return r26906718;
}
↓
double f(double x, double y, double z, double t) {
        double r26906719 = x;
        double r26906720 = 1.0;
        double r26906721 = z;
        double r26906722 = t;
        double r26906723 = r26906722 / r26906721;
        double r26906724 = y;
        double r26906725 = 2.0;
        double r26906726 = r26906724 / r26906725;
        double r26906727 = r26906723 * r26906726;
        double r26906728 = r26906721 - r26906727;
        double r26906729 = r26906720 / r26906728;
        double r26906730 = r26906729 * r26906724;
        double r26906731 = r26906719 - r26906730;
        return r26906731;
}

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
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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

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

  4. Applied div-inv1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2019172 +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)))))