Average Error: 0.1 → 0.1

Time: 7.5s

Precision: binary64

Cost: 832

\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]

↓

\[\left(x \cdot y + z \cdot z\right) + z \cdot \left(z + z\right)\]

\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z

↓

\left(x \cdot y + z \cdot z\right) + z \cdot \left(z + z\right)

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))

↓

(FPCore (x y z) :precision binary64 (+ (+ (* x y) (* z z)) (* z (+ z z))))

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * z);
}

↓

double code(double x, double y, double z) {
	return ((x * y) + (z * z)) + (z * (z + 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	0.1
Target	0.1
Herbie	0.1

\[\left(3 \cdot z\right) \cdot z + y \cdot x\]

Alternatives

Alternative 1
Error	0.3
Cost	45888

\[x \cdot y + \sqrt{z \cdot \left(z \cdot 3\right)} \cdot \left(\sqrt{\left|z \cdot \sqrt{3}\right|} \cdot \sqrt{\left|z \cdot \sqrt{3}\right|}\right)\]

Alternative 2
Error	1.2
Cost	21312

\[z \cdot z + \sqrt[3]{x \cdot y + z \cdot \left(z + z\right)} \cdot \left(\sqrt[3]{x \cdot y + z \cdot \left(z + z\right)} \cdot \sqrt[3]{x \cdot y + z \cdot \left(z + z\right)}\right)\]

Alternative 3
Error	1.3
Cost	21056

\[\sqrt[3]{x \cdot y + \left(z \cdot z\right) \cdot 3} \cdot \left(\sqrt[3]{x \cdot y + \left(z \cdot z\right) \cdot 3} \cdot \sqrt[3]{x \cdot y + \left(z \cdot z\right) \cdot 3}\right)\]

Alternative 4
Error	0.5
Cost	20544

\[x \cdot y + \sqrt[3]{z \cdot \left(z \cdot 3\right)} \cdot \left(\sqrt[3]{z \cdot \left(z \cdot 3\right)} \cdot \sqrt[3]{z \cdot \left(z \cdot 3\right)}\right)\]

Alternative 5
Error	31.9
Cost	20160

\[x \cdot y + \sqrt{z \cdot \left(z \cdot 3\right)} \cdot \left(\sqrt{z} \cdot \sqrt{z \cdot 3}\right)\]

Alternative 6
Error	0.2
Cost	20032

\[x \cdot y + \sqrt{z \cdot \left(z \cdot 3\right)} \cdot \left|z \cdot \sqrt{3}\right|\]

Alternative 7
Error	12.6
Cost	19904

\[x \cdot y - \left(z \cdot \sqrt{3}\right) \cdot \left|z \cdot \sqrt{3}\right|\]

Alternative 8
Error	18.9
Cost	14272

\[z \cdot z + \sqrt{x \cdot y + z \cdot \left(z + z\right)} \cdot \sqrt{x \cdot y + z \cdot \left(z + z\right)}\]

Alternative 9
Error	19.0
Cost	14016

\[\sqrt{x \cdot y + \left(z \cdot z\right) \cdot 3} \cdot \sqrt{x \cdot y + \left(z \cdot z\right) \cdot 3}\]

Alternative 10
Error	0.2
Cost	13760

\[x \cdot y + \sqrt{z \cdot \left(z \cdot 3\right)} \cdot \sqrt{z \cdot \left(z \cdot 3\right)}\]

Alternative 11
Error	12.6
Cost	13632

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

Alternative 12
Error	12.0
Cost	13632

\[x \cdot y + \sqrt{z \cdot \left(z \cdot 3\right)} \cdot \left(z \cdot \sqrt{3}\right)\]

Alternative 13
Error	0.3
Cost	13504

\[x \cdot y + \left(z \cdot \sqrt{3}\right) \cdot \left(z \cdot \sqrt{3}\right)\]

Alternative 14
Error	31.9
Cost	13440

\[x \cdot y + \sqrt{z} \cdot \left(3 \cdot {z}^{1.5}\right)\]

Alternative 15
Error	0.2
Cost	13440

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

Alternative 16
Error	13.3
Cost	13376

\[x \cdot y + {\left(9 \cdot {z}^{4}\right)}^{0.5}\]

Alternative 17
Error	13.3
Cost	13312

\[x \cdot y + \sqrt{9 \cdot {z}^{4}}\]

Alternative 18
Error	27.2
Cost	1344

\[z \cdot z + \frac{\left(x \cdot y\right) \cdot \left(x \cdot y + z \cdot \left(z + z\right)\right)}{x \cdot y}\]

Alternative 19
Error	0.1
Cost	960

\[z \cdot z + \left(z \cdot z + \left(x \cdot y + z \cdot z\right)\right)\]

Alternative 20
Error	0.1
Cost	576

\[x \cdot y + z \cdot \left(z \cdot 3\right)\]

Alternative 21
Error	35.9
Cost	320

\[\left(z \cdot z\right) \cdot 3\]

Alternative 22
Error	23.9
Cost	192

\[x \cdot y\]

Alternative 23
Error	61.3
Cost	64

\[1\]

Alternative 24
Error	59.2
Cost	64

\[0\]

Alternative 25
Error	62.2
Cost	64

\[-1\]

Error

Derivation

Initial program 0.1
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
Using strategy rm
Applied associate-+l+_binary64_106600.1
\[\leadsto \color{blue}{\left(x \cdot y + z \cdot z\right) + \left(z \cdot z + z \cdot z\right)}\]
Simplified0.1
\[\leadsto \left(x \cdot y + z \cdot z\right) + \color{blue}{z \cdot \left(z + z\right)}\]
Simplified0.1
\[\leadsto \color{blue}{\left(x \cdot y + z \cdot z\right) + z \cdot \left(z + z\right)}\]
Final simplification0.1
\[\leadsto \left(x \cdot y + z \cdot z\right) + z \cdot \left(z + z\right)\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))