Average Error: 11.5 → 1.1

Time: 9.3s

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

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	11.5
Target	0.1
Herbie	1.1

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

Alternatives

Alternative 1
Error	3.0
Cost	40768

\[x - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}} \cdot \sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}}\]

Alternative 2
Error	43.4
Cost	27328

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

Alternative 3
Error	3.0
Cost	21440

\[x - \frac{1}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}} \cdot \sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}} \cdot \frac{y}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}}\]

Alternative 4
Error	3.0
Cost	21312

\[x - \frac{\frac{y}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}} \cdot \sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}}}{\sqrt[3]{z - \frac{t \cdot y}{z \cdot 2}}}\]

Alternative 5
Error	33.0
Cost	14528

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

Alternative 6
Error	33.6
Cost	14272

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

Alternative 7
Error	1.2
Cost	13888

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

Alternative 8
Error	32.9
Cost	13760

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

Alternative 9
Error	33.4
Cost	13760

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

Alternative 10
Error	8.7
Cost	13696

\[x - \sqrt[3]{{\left(\frac{y}{z - \frac{t \cdot y}{z \cdot 2}}\right)}^{3}}\]

Alternative 11
Error	23.7
Cost	13632

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

Alternative 12
Error	19.4
Cost	1984

\[x - \frac{y}{z \cdot z - \left(y \cdot \frac{t}{z}\right) \cdot \frac{t \cdot y}{z \cdot 4}} \cdot \left(z + \frac{t \cdot y}{z \cdot 2}\right)\]

Alternative 13
Error	11.5
Cost	1088

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

Alternative 14
Error	2.8
Cost	960

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

Alternative 15
Error	2.8
Cost	960

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

Alternative 16
Error	2.8
Cost	960

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

Alternative 17
Error	1.1
Cost	832

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

Alternative 18
Error	2.7
Cost	832

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

Alternative 19
Error	49.6
Cost	768

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

Alternative 20
Error	48.2
Cost	768

\[\frac{-y}{z - \left(y \cdot \frac{t}{z}\right) \cdot 0.5}\]

Alternative 21
Error	24.0
Cost	448

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

Alternative 22
Error	23.9
Cost	320

\[x - \frac{y}{z}\]

Alternative 23
Error	15.5
Cost	64

\[x\]

Alternative 24
Error	61.6
Cost	64

\[1\]

Alternative 25
Error	61.9
Cost	64

\[0\]

Alternative 26
Error	61.7
Cost	64

\[-1\]

Error

Derivation

Initial program 11.5
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
Simplified2.7
\[\leadsto \color{blue}{x - \frac{y}{z - \frac{y \cdot t}{2 \cdot z}}}\]
Using strategy rm
Applied associate-/l*_binary64_167321.1
\[\leadsto x - \frac{y}{z - \color{blue}{\frac{y}{\frac{2 \cdot z}{t}}}}\]
Simplified1.1
\[\leadsto x - \frac{y}{z - \frac{y}{\color{blue}{\frac{2}{\frac{t}{z}}}}}\]
Simplified1.1
\[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{\frac{2}{\frac{t}{z}}}}}\]
Final simplification1.1
\[\leadsto x - \frac{y}{z - \frac{y}{\frac{2}{\frac{t}{z}}}}\]

Reproduce

herbie shell --seed 2021022 
(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)))))