Average Error: 11.8 → 1.1
Time: 6.8s
Precision: binary64
Cost: 832
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \frac{y}{\frac{2}{\frac{t}{z}}}}\]
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{2}{\frac{t}{z}}}}
(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 (/ 2.0 (/ t z)))))))
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 / (2.0 / (t / z)))));
}

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

Alternatives

Alternative 1
Error12.2
Cost26240
\[x - \frac{y}{z - \log \left({\left(\sqrt{e^{\frac{y}{z}}}\right)}^{t}\right)}\]
Alternative 2
Error2.9
Cost21056
\[x - \frac{y}{z - \sqrt[3]{\frac{y \cdot t}{z \cdot 2}} \cdot \left(\sqrt[3]{\frac{y \cdot t}{z \cdot 2}} \cdot \sqrt[3]{\frac{y \cdot t}{z \cdot 2}}\right)}\]
Alternative 3
Error3.1
Cost20288
\[x - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{\sqrt[3]{y}}{z - \frac{y \cdot t}{z \cdot 2}}\]
Alternative 4
Error5.7
Cost13696
\[x - \frac{y}{z - \sqrt[3]{\frac{{\left(y \cdot \frac{t}{z}\right)}^{3}}{8}}}\]
Alternative 5
Error11.8
Cost1088
\[x - \frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t}\]
Alternative 6
Error2.8
Cost960
\[x - \frac{y}{z - \frac{1}{\frac{z \cdot 2}{y \cdot t}}}\]
Alternative 7
Error1.1
Cost832
\[x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2}}\]
Alternative 8
Error2.8
Cost832
\[x - \frac{y}{z - \frac{y \cdot t}{z \cdot 2}}\]
Alternative 9
Error24.3
Cost448
\[x - -2 \cdot \frac{z}{t}\]
Alternative 10
Error54.9
Cost320
\[z \cdot \frac{2}{t}\]
Alternative 11
Error23.3
Cost320
\[x - \frac{y}{z}\]
Alternative 12
Error15.8
Cost64
\[x\]
Alternative 13
Error61.7
Cost64
\[1\]
Alternative 14
Error61.9
Cost64
\[0\]
Alternative 15
Error61.6
Cost64
\[-1\]

Error

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. Simplified2.8

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

  4. Applied associate-/l*_binary64_116711.1

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

  5. Simplified1.1

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

  6. Simplified1.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2021042 
(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)))))