Average Error: 11.5 → 0.1
Time: 17.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r360485 = x;
        double r360486 = y;
        double r360487 = 2.0;
        double r360488 = r360486 * r360487;
        double r360489 = z;
        double r360490 = r360488 * r360489;
        double r360491 = r360489 * r360487;
        double r360492 = r360491 * r360489;
        double r360493 = t;
        double r360494 = r360486 * r360493;
        double r360495 = r360492 - r360494;
        double r360496 = r360490 / r360495;
        double r360497 = r360485 - r360496;
        return r360497;
}
↓
double f(double x, double y, double z, double t) {
        double r360498 = x;
        double r360499 = 1.0;
        double r360500 = z;
        double r360501 = y;
        double r360502 = r360500 / r360501;
        double r360503 = 0.5;
        double r360504 = t;
        double r360505 = r360504 / r360500;
        double r360506 = r360503 * r360505;
        double r360507 = r360502 - r360506;
        double r360508 = r360499 / r360507;
        double r360509 = r360498 - r360508;
        return r360509;
}

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

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. Simplified0.9

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

  4. Applied clear-num0.9

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

  5. Simplified2.9

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

  6. Taylor expanded around 0 0.1

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

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2019325 +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)))))