Average Error: 11.6 → 1.2
Time: 23.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - y \cdot \frac{1}{\frac{t}{z} \cdot \left(-\frac{y}{2}\right) + z}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - y \cdot \frac{1}{\frac{t}{z} \cdot \left(-\frac{y}{2}\right) + z}
double f(double x, double y, double z, double t) {
        double r357529 = x;
        double r357530 = y;
        double r357531 = 2.0;
        double r357532 = r357530 * r357531;
        double r357533 = z;
        double r357534 = r357532 * r357533;
        double r357535 = r357533 * r357531;
        double r357536 = r357535 * r357533;
        double r357537 = t;
        double r357538 = r357530 * r357537;
        double r357539 = r357536 - r357538;
        double r357540 = r357534 / r357539;
        double r357541 = r357529 - r357540;
        return r357541;
}
↓
double f(double x, double y, double z, double t) {
        double r357542 = x;
        double r357543 = y;
        double r357544 = 1.0;
        double r357545 = t;
        double r357546 = z;
        double r357547 = r357545 / r357546;
        double r357548 = 2.0;
        double r357549 = r357543 / r357548;
        double r357550 = -r357549;
        double r357551 = r357547 * r357550;
        double r357552 = r357551 + r357546;
        double r357553 = r357544 / r357552;
        double r357554 = r357543 * r357553;
        double r357555 = r357542 - r357554;
        return r357555;
}

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

Original11.6
Target0.1
Herbie1.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified1.1

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

  4. Applied div-inv1.2

    \[\leadsto x - \color{blue}{y \cdot \frac{1}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}}\]
  5. Using strategy rm

  6. Applied fma-udef1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019323 +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)))))