Average Error: 12.1 → 1.0
Time: 15.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - 2 \cdot \frac{y}{2 \cdot z - \frac{y}{\frac{z}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - 2 \cdot \frac{y}{2 \cdot z - \frac{y}{\frac{z}{t}}}
double f(double x, double y, double z, double t) {
        double r388475 = x;
        double r388476 = y;
        double r388477 = 2.0;
        double r388478 = r388476 * r388477;
        double r388479 = z;
        double r388480 = r388478 * r388479;
        double r388481 = r388479 * r388477;
        double r388482 = r388481 * r388479;
        double r388483 = t;
        double r388484 = r388476 * r388483;
        double r388485 = r388482 - r388484;
        double r388486 = r388480 / r388485;
        double r388487 = r388475 - r388486;
        return r388487;
}
↓
double f(double x, double y, double z, double t) {
        double r388488 = x;
        double r388489 = 2.0;
        double r388490 = y;
        double r388491 = z;
        double r388492 = r388489 * r388491;
        double r388493 = t;
        double r388494 = r388491 / r388493;
        double r388495 = r388490 / r388494;
        double r388496 = r388492 - r388495;
        double r388497 = r388490 / r388496;
        double r388498 = r388489 * r388497;
        double r388499 = r388488 - r388498;
        return r388499;
}

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

Original12.1
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  2. Simplified2.8

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

  4. Applied associate-/l*1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2019196 
(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)))))