Average Error: 11.6 → 1.0
Time: 3.5min
Precision: binary64
Cost: 832
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z + \frac{y}{\frac{z \cdot -2}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z + \frac{y}{\frac{z \cdot -2}{t}}}
(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))
↓
(FPCore (x y z t) :precision binary64 (- x (/ y (+ z (/ y (/ (* z -2.0) t))))))
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 x - (y / (z + (y / ((z * -2.0) / t))));
}

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
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]
Alternative 1
Accuracy1.1
Cost832
\[x - \frac{y}{z - y \cdot \frac{0.5}{\frac{z}{t}}}\]
Alternative 2
Accuracy2.8
Cost832
\[x + \frac{y}{\frac{y \cdot t}{z \cdot 2} - z}\]

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. Simplified2.8

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

  4. Applied associate-/l*_binary64_129811.0

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

  5. Simplified1.1

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

  6. Taylor expanded around 0 1.0

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

  8. Applied frac-2neg_binary64_130471.0

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

  9. Simplified1.0

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

  11. Applied associate-*l/_binary64_129791.0

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))