Average Error: 11.6 → 2.2
Time: 4.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\mathsf{fma}\left(1, x, \frac{2}{2 \cdot z - \frac{t}{\frac{z}{y}}} \cdot \left(-y\right)\right)\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\mathsf{fma}\left(1, x, \frac{2}{2 \cdot z - \frac{t}{\frac{z}{y}}} \cdot \left(-y\right)\right)
double code(double x, double y, double z, double t) {
	return (x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t))));
}
↓
double code(double x, double y, double z, double t) {
	return fma(1.0, x, ((2.0 / ((2.0 * z) - (t / (z / y)))) * -y));
}

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
Herbie2.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Using strategy rm

  3. Applied associate-/l*6.5

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

  5. Applied *-un-lft-identity6.5

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

  6. Applied *-un-lft-identity6.5

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

  7. Applied times-frac6.5

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

  8. Simplified6.5

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

  9. Simplified2.7

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

  11. Applied associate-/l*2.2

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

  13. Applied *-un-lft-identity2.2

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

  14. Applied fma-neg2.2

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

  15. Simplified2.2

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

  16. Final simplification2.2

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

Reproduce

herbie shell --seed 2020075 +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)))))