Average Error: 11.3 → 0.9
Time: 12.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\frac{\left(-2\right) \cdot y}{z \cdot 2 - \frac{y}{\frac{z}{t}}} + x\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\frac{\left(-2\right) \cdot y}{z \cdot 2 - \frac{y}{\frac{z}{t}}} + x
double f(double x, double y, double z, double t) {
        double r261279 = x;
        double r261280 = y;
        double r261281 = 2.0;
        double r261282 = r261280 * r261281;
        double r261283 = z;
        double r261284 = r261282 * r261283;
        double r261285 = r261283 * r261281;
        double r261286 = r261285 * r261283;
        double r261287 = t;
        double r261288 = r261280 * r261287;
        double r261289 = r261286 - r261288;
        double r261290 = r261284 / r261289;
        double r261291 = r261279 - r261290;
        return r261291;
}
↓
double f(double x, double y, double z, double t) {
        double r261292 = 2.0;
        double r261293 = -r261292;
        double r261294 = y;
        double r261295 = r261293 * r261294;
        double r261296 = z;
        double r261297 = r261296 * r261292;
        double r261298 = t;
        double r261299 = r261296 / r261298;
        double r261300 = r261294 / r261299;
        double r261301 = r261297 - r261300;
        double r261302 = r261295 / r261301;
        double r261303 = x;
        double r261304 = r261302 + r261303;
        return r261304;
}

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

Original11.3
Target0.1
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Simplified2.6

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

  4. Applied associate-/l*0.9

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

  6. Applied fma-udef0.9

    \[\leadsto \color{blue}{\left(-\frac{2}{z \cdot 2 - \frac{y}{\frac{z}{t}}}\right) \cdot y + x}\]
  7. Using strategy rm

  8. Applied distribute-neg-frac0.9

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

  9. Applied associate-*l/0.9

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

  10. Final simplification0.9

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

Reproduce

herbie shell --seed 2019208 +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)))))