Average Error: 11.5 → 1.1
Time: 16.8s
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 r23933460 = x;
        double r23933461 = y;
        double r23933462 = 2.0;
        double r23933463 = r23933461 * r23933462;
        double r23933464 = z;
        double r23933465 = r23933463 * r23933464;
        double r23933466 = r23933464 * r23933462;
        double r23933467 = r23933466 * r23933464;
        double r23933468 = t;
        double r23933469 = r23933461 * r23933468;
        double r23933470 = r23933467 - r23933469;
        double r23933471 = r23933465 / r23933470;
        double r23933472 = r23933460 - r23933471;
        return r23933472;
}
↓
double f(double x, double y, double z, double t) {
        double r23933473 = x;
        double r23933474 = 1.0;
        double r23933475 = z;
        double r23933476 = t;
        double r23933477 = r23933476 / r23933475;
        double r23933478 = y;
        double r23933479 = 2.0;
        double r23933480 = r23933478 / r23933479;
        double r23933481 = r23933477 * r23933480;
        double r23933482 = r23933475 - r23933481;
        double r23933483 = r23933474 / r23933482;
        double r23933484 = r23933483 * r23933478;
        double r23933485 = r23933473 - r23933484;
        return r23933485;
}

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)))))