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

Average Accuracy: 68.9% → 82.7%

Time: 11.6s

Precision: binary64

Cost: 13380

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-243}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e-243)
   (* 2.0 (sqrt (* (+ y z) x)))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

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) {
	double tmp;
	if (y <= 1.15e-243) {
		tmp = 2.0 * sqrt(((y + z) * x));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 1.15d-243) then
        tmp = 2.0d0 * sqrt(((y + z) * x))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    end if
    code = tmp
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) {
	double tmp;
	if (y <= 1.15e-243) {
		tmp = 2.0 * Math.sqrt(((y + z) * x));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= 1.15e-243:
		tmp = 2.0 * math.sqrt(((y + z) * x))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	return tmp

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)
	tmp = 0.0
	if (y <= 1.15e-243)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y + z) * x)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e-243)
		tmp = 2.0 * sqrt(((y + z) * x));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[y, 1.15e-243], N[(2.0 * N[Sqrt[N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	68.9%
Target	82.2%
Herbie	82.7%

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

Derivation

1. Split input into 2 regimes

2. if y < 1.15e-243

   1. Initial program 69.1%

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

   2. Simplified 69.1%

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

      [Start] 69.1	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.1	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 67.3%

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

   if 1.15e-243 < y

   1. Initial program 68.8%

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

   2. Simplified 68.8%

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

      [Start] 68.8	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 68.8	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 68.8%

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

      [Start] 68.8	\[ 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z} \]
      add-cube-cbrt [=>] 68.8	\[ 2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{x \cdot \left(y + z\right)} \cdot \sqrt[3]{x \cdot \left(y + z\right)}\right) \cdot \sqrt[3]{x \cdot \left(y + z\right)}} + y \cdot z} \]
      pow3 [=>] 68.8	\[ 2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{x \cdot \left(y + z\right)}\right)}^{3}} + y \cdot z} \]

   4. Taylor expanded in y around 0 34.7%

      \[\leadsto 2 \cdot \sqrt{{\color{blue}{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot z} \]

   5. Applied egg-rr 68.0%

      \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{z \cdot \left(x + y\right)}\right)}^{1.5}} \]
      Proof

      [Start] 34.7	\[ 2 \cdot \sqrt{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z} \]
      rem-cbrt-cube [<=] 34.7	\[ 2 \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3}}\right)}}^{3} + y \cdot z} \]
      unpow1/3 [<=] 34.7	\[ 2 \cdot \sqrt{{\color{blue}{\left({\left({\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot z} \]
      unpow1/3 [=>] 34.7	\[ 2 \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3}}\right)}}^{3} + y \cdot z} \]
      rem-cube-cbrt [=>] 34.7	\[ 2 \cdot \sqrt{\color{blue}{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3}} + y \cdot z} \]
      pow1/2 [=>] 34.7	\[ 2 \cdot \color{blue}{{\left({\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z\right)}^{0.5}} \]
      add-cube-cbrt [=>] 34.4	\[ 2 \cdot {\color{blue}{\left(\left(\sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z} \cdot \sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z}\right) \cdot \sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z}\right)}}^{0.5} \]
      pow3 [=>] 34.4	\[ 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z}\right)}^{3}\right)}}^{0.5} \]
      pow-pow [=>] 34.4	\[ 2 \cdot \color{blue}{{\left(\sqrt[3]{{\left({\left(z \cdot x\right)}^{0.3333333333333333}\right)}^{3} + y \cdot z}\right)}^{\left(3 \cdot 0.5\right)}} \]
      pow-pow [=>] 68.0	\[ 2 \cdot {\left(\sqrt[3]{\color{blue}{{\left(z \cdot x\right)}^{\left(0.3333333333333333 \cdot 3\right)}} + y \cdot z}\right)}^{\left(3 \cdot 0.5\right)} \]
      metadata-eval [=>] 68.0	\[ 2 \cdot {\left(\sqrt[3]{{\left(z \cdot x\right)}^{\color{blue}{1}} + y \cdot z}\right)}^{\left(3 \cdot 0.5\right)} \]
      pow1 [<=] 68.0	\[ 2 \cdot {\left(\sqrt[3]{\color{blue}{z \cdot x} + y \cdot z}\right)}^{\left(3 \cdot 0.5\right)} \]
      *-commutative [=>] 68.0	\[ 2 \cdot {\left(\sqrt[3]{z \cdot x + \color{blue}{z \cdot y}}\right)}^{\left(3 \cdot 0.5\right)} \]
      distribute-lft-out [=>] 68.0	\[ 2 \cdot {\left(\sqrt[3]{\color{blue}{z \cdot \left(x + y\right)}}\right)}^{\left(3 \cdot 0.5\right)} \]
      metadata-eval [=>] 68.0	\[ 2 \cdot {\left(\sqrt[3]{z \cdot \left(x + y\right)}\right)}^{\color{blue}{1.5}} \]

   6. Applied egg-rr 99.4%

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

      [Start] 68.0	\[ 2 \cdot {\left(\sqrt[3]{z \cdot \left(x + y\right)}\right)}^{1.5} \]
      pow1/3 [=>] 63.0	\[ 2 \cdot {\color{blue}{\left({\left(z \cdot \left(x + y\right)\right)}^{0.3333333333333333}\right)}}^{1.5} \]
      pow-pow [=>] 68.7	\[ 2 \cdot \color{blue}{{\left(z \cdot \left(x + y\right)\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]
      metadata-eval [=>] 68.7	\[ 2 \cdot {\left(z \cdot \left(x + y\right)\right)}^{\color{blue}{0.5}} \]
      pow1/2 [<=] 68.7	\[ 2 \cdot \color{blue}{\sqrt{z \cdot \left(x + y\right)}} \]
      *-commutative [=>] 68.7	\[ 2 \cdot \sqrt{\color{blue}{\left(x + y\right) \cdot z}} \]
      sqrt-prod [=>] 99.4	\[ 2 \cdot \color{blue}{\left(\sqrt{x + y} \cdot \sqrt{z}\right)} \]

   7. Simplified 99.4%

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

      [Start] 99.4	\[ 2 \cdot \left(\sqrt{x + y} \cdot \sqrt{z}\right) \]
      +-commutative [<=] 99.4	\[ 2 \cdot \left(\sqrt{\color{blue}{y + x}} \cdot \sqrt{z}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-243}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.4%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-243}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	68.9%
Cost	7104

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

Alternative 3
Accuracy	67.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Accuracy	68.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-277}:\\ \;\;\;\;2 \cdot \sqrt{\left(y + z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Accuracy	66.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 6
Accuracy	35.1%
Cost	6720

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

Reproduce

herbie shell --seed 2023142 
(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)))))