Average Error: 11.3 → 1.0
Time: 22.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \frac{y}{\left(2 \cdot z\right) \cdot \frac{1}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \frac{y}{\left(2 \cdot z\right) \cdot \frac{1}{t}}}
double f(double x, double y, double z, double t) {
        double r1234263 = x;
        double r1234264 = y;
        double r1234265 = 2.0;
        double r1234266 = r1234264 * r1234265;
        double r1234267 = z;
        double r1234268 = r1234266 * r1234267;
        double r1234269 = r1234267 * r1234265;
        double r1234270 = r1234269 * r1234267;
        double r1234271 = t;
        double r1234272 = r1234264 * r1234271;
        double r1234273 = r1234270 - r1234272;
        double r1234274 = r1234268 / r1234273;
        double r1234275 = r1234263 - r1234274;
        return r1234275;
}
↓
double f(double x, double y, double z, double t) {
        double r1234276 = x;
        double r1234277 = y;
        double r1234278 = z;
        double r1234279 = 2.0;
        double r1234280 = r1234279 * r1234278;
        double r1234281 = 1.0;
        double r1234282 = t;
        double r1234283 = r1234281 / r1234282;
        double r1234284 = r1234280 * r1234283;
        double r1234285 = r1234277 / r1234284;
        double r1234286 = r1234278 - r1234285;
        double r1234287 = r1234277 / r1234286;
        double r1234288 = r1234276 - r1234287;
        return r1234288;
}

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

Enter valid numbers for all inputs

Target

Original11.3
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified2.7

    \[\leadsto \color{blue}{x - \frac{y}{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}}\]
  3. Using strategy rm

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

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

  5. Applied times-frac1.0

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

  6. Applied associate-/l*1.0

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

  7. Taylor expanded around 0 1.0

    \[\leadsto x - \frac{y}{\frac{z}{1} - \frac{\frac{y}{1}}{\color{blue}{2 \cdot \frac{z}{t}}}}\]
  8. Using strategy rm

  9. Applied div-inv1.0

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

  10. Applied associate-*r*1.0

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

  11. Final simplification1.0

    \[\leadsto x - \frac{y}{z - \frac{y}{\left(2 \cdot z\right) \cdot \frac{1}{t}}}\]

Reproduce

herbie shell --seed 2019198 
(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)))))