Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5

Average Error: 19.9 → 19.9

Time: 15.9s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

(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 x))))))

C

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 + x))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt(((x * y) + (z * (y + x))))
end function

Java

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

↓

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

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

↓

def code(x, y, z):
	return 2.0 * math.sqrt(((x * y) + (z * (y + x))))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

↓

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(x * y) + Float64(z * Float64(y + x)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

↓

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt(((x * y) + (z * (y + x))));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(x * y), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	19.9
Target	19.3
Herbie	19.9

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

Derivation

1. Initial program 19.9

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

2. Simplified 19.9

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

   [Start] 19.9	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
   rational.json-simplify-1 [=>] 19.9	\[ 2 \cdot \sqrt{\color{blue}{y \cdot z + \left(x \cdot y + x \cdot z\right)}} \]
   rational.json-simplify-41 [=>] 19.9	\[ 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]
   rational.json-simplify-2 [=>] 19.9	\[ 2 \cdot \sqrt{x \cdot y + \left(\color{blue}{z \cdot x} + y \cdot z\right)} \]
   rational.json-simplify-51 [=>] 19.9	\[ 2 \cdot \sqrt{x \cdot y + \color{blue}{z \cdot \left(y + x\right)}} \]

3. Final simplification 19.9

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

Alternatives

Alternative 1
Error	42.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \]

Alternative 2
Error	33.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + x\right) \cdot z}\\ \end{array} \]

Alternative 3
Error	48.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 4
Error	48.9
Cost	6720

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

Reproduce

herbie shell --seed 2023068 
(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)))))