Average Error: 11.2 → 0.1
Time: 14.0s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]
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} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}
double f(double x, double y, double z, double t) {
        double r19903052 = x;
        double r19903053 = y;
        double r19903054 = 2.0;
        double r19903055 = r19903053 * r19903054;
        double r19903056 = z;
        double r19903057 = r19903055 * r19903056;
        double r19903058 = r19903056 * r19903054;
        double r19903059 = r19903058 * r19903056;
        double r19903060 = t;
        double r19903061 = r19903053 * r19903060;
        double r19903062 = r19903059 - r19903061;
        double r19903063 = r19903057 / r19903062;
        double r19903064 = r19903052 - r19903063;
        return r19903064;
}
↓
double f(double x, double y, double z, double t) {
        double r19903065 = x;
        double r19903066 = 1.0;
        double r19903067 = z;
        double r19903068 = y;
        double r19903069 = r19903067 / r19903068;
        double r19903070 = 0.5;
        double r19903071 = t;
        double r19903072 = r19903070 * r19903071;
        double r19903073 = r19903066 / r19903067;
        double r19903074 = r19903072 * r19903073;
        double r19903075 = r19903069 - r19903074;
        double r19903076 = r19903066 / r19903075;
        double r19903077 = r19903065 - r19903076;
        return r19903077;
}

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

Original11.2
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.2

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

  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. Using strategy rm

  7. Applied div-inv0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]

Reproduce

herbie shell --seed 2019162 
(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)))))