Average Error: 20.1 → 20.1

Time: 6.9s

Precision: binary64

Cost: 7104

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

↓

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

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

↓

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

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

↓

(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (+ (* x (+ y z)) (* y z)))))

double code(double x, double y, double z) {
	return 2.0 * sqrt(((x * y) + (x * z)) + (y * z));
}

↓

double code(double x, double y, double z) {
	return 2.0 * sqrt((x * (y + z)) + (y * 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	20.1
Target	19.4
Herbie	20.1

\[\begin{array}{l} \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Alternatives

Alternative 1
Error	50.5
Cost	92864

\[2 \cdot \left(\sqrt{\sqrt{x \cdot \left(y + z\right) + y \cdot z}} \cdot \sqrt{\sqrt{\sqrt[3]{x \cdot \left(y + z\right) + y \cdot z}} \cdot \left|\sqrt[3]{e^{\log \left(-\left(x + z\right)\right) - \log \left(\frac{-1}{y}\right)}} + 0.3333333333333333 \cdot \frac{\sqrt[3]{e^{\log \left(-\left(x + z\right)\right) - \log \left(\frac{-1}{y}\right)}}}{\frac{y}{\frac{x \cdot z}{x + z}}}\right|}\right)\]

Alternative 2
Error	20.5
Cost	60224

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

Alternative 3
Error	22.3
Cost	59584

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

Alternative 4
Error	21.7
Cost	59584

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

Alternative 5
Error	20.5
Cost	46784

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

Alternative 6
Error	20.5
Cost	46784

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

Alternative 7
Error	48.9
Cost	46400

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

Alternative 8
Error	48.1
Cost	46400

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

Alternative 9
Error	20.9
Cost	40384

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

Alternative 10
Error	22.6
Cost	39744

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

Alternative 11
Error	20.6
Cost	26944

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

Alternative 12
Error	20.4
Cost	26944

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

Alternative 13
Error	20.4
Cost	26944

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

Alternative 14
Error	49.2
Cost	26560

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

Alternative 15
Error	48.4
Cost	26560

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

Alternative 16
Error	52.3
Cost	20288

\[2 \cdot \left(\sqrt{x \cdot y} + z \cdot \left(0.5 \cdot \left(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\right)\right)\right)\]

Alternative 17
Error	51.5
Cost	20288

\[2 \cdot \left(\sqrt{x \cdot z} + \left(y \cdot 0.5\right) \cdot \left(\sqrt{\frac{x}{z}} + \sqrt{\frac{z}{x}}\right)\right)\]

Alternative 18
Error	51.6
Cost	20288

\[2 \cdot \left(\sqrt{y \cdot z} + x \cdot \left(0.5 \cdot \left(\sqrt{\frac{y}{z}} + \sqrt{\frac{z}{y}}\right)\right)\right)\]

Alternative 19
Error	32.7
Cost	19968

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

Alternative 20
Error	48.1
Cost	14784

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

Alternative 21
Error	39.7
Cost	8384

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

Alternative 22
Error	20.1
Cost	7232

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

Alternative 23
Error	20.1
Cost	7104

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

Alternative 24
Error	34.0
Cost	6848

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

Alternative 25
Error	34.8
Cost	6848

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

Alternative 26
Error	34.7
Cost	6848

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

Alternative 27
Error	49.1
Cost	6720

\[2 \cdot \sqrt{x \cdot y}\]

Alternative 28
Error	48.3
Cost	6720

\[2 \cdot \sqrt{x \cdot z}\]

Alternative 29
Error	48.2
Cost	6720

\[2 \cdot \sqrt{y \cdot z}\]

Alternative 30
Error	64.0
Cost	448

\[2 \cdot \left(\left(y + z\right) \cdot \mathsf{NaN}\right)\]

Alternative 31
Error	60.1
Cost	64

\[1\]

Alternative 32
Error	62.2
Cost	64

\[0\]

Alternative 33
Error	63.1
Cost	64

\[-1\]

Error

Derivation

Initial program 20.1
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
Simplified20.1
\[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}}\]
Simplified20.1
\[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}}\]
Final simplification20.1
\[\leadsto 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))