Average Error: 11.8 → 0.1
Time: 4.3s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]
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} - \frac{0.5}{\frac{z}{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 (/ 1.0 (- (/ z y) (/ 0.5 (/ z 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 - (1.0 / ((z / y) - (0.5 / (z / 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

Original11.8
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.8

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

  2. Simplified2.9

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

  4. Applied associate-/l*_binary641.0

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

  5. Simplified1.0

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

  7. Applied clear-num_binary641.1

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

  8. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity_binary640.1

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

  11. Applied *-un-lft-identity_binary640.1

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

  12. Applied times-frac_binary640.1

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

  13. Applied *-un-lft-identity_binary640.1

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

  14. Applied *-un-lft-identity_binary640.1

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

  15. Applied times-frac_binary640.1

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

  16. Applied distribute-lft-out--_binary640.1

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

  17. Final simplification0.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))