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

Percentage Accurate: 70.2% → 95.6%

Time: 8.6s

Precision: binary64

Cost: 26436

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

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

↓

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+53)
   (* 2.0 (pow (exp (* 0.25 (- (log (- x)) (log (/ -1.0 y))))) 2.0))
   (if (<= y 6.5e-257)
     (* 2.0 (sqrt (+ (* x z) (* y (+ x z)))))
     (* 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)));
}

↓

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+53) {
		tmp = 2.0 * pow(exp((0.25 * (log(-x) - log((-1.0 / y))))), 2.0);
	} else if (y <= 6.5e-257) {
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	} 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

↓

NOTE: x, y, and z should be sorted in increasing order before calling this 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 <= (-7.2d+53)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log(-x) - log(((-1.0d0) / y))))) ** 2.0d0)
    else if (y <= 6.5d-257) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * (x + z))))
    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)));
}

↓

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+53) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(-x) - Math.log((-1.0 / y))))), 2.0);
	} else if (y <= 6.5e-257) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * (x + z))));
	} 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)))

↓

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= -7.2e+53:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log(-x) - math.log((-1.0 / y))))), 2.0)
	elif y <= 6.5e-257:
		tmp = 2.0 * math.sqrt(((x * z) + (y * (x + z))))
	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

↓

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+53)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	elseif (y <= 6.5e-257)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * Float64(x + z)))));
	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

↓

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.2e+53)
		tmp = 2.0 * (exp((0.25 * (log(-x) - log((-1.0 / y))))) ^ 2.0);
	elseif (y <= 6.5e-257)
		tmp = 2.0 * sqrt(((x * z) + (y * (x + z))));
	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]

↓

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -7.2e+53], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-257], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	70.2%
Target	82.7%
Herbie	95.6%

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

Derivation

1. Split input into 3 regimes

2. if y < -7.2e53

   1. Initial program 58.9%

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

   2. Simplified 58.9%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}} \]
      Step-by-step derivation

      [Start] 58.9%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 58.9%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 58.7%

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25}\right)}^{2}} \]
      Step-by-step derivation

      [Start] 58.9%	\[ 2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z} \]
      flip-+ [=>] 12.6%	\[ 2 \cdot \sqrt{\color{blue}{\frac{\left(x \cdot \left(y + z\right)\right) \cdot \left(x \cdot \left(y + z\right)\right) - \left(y \cdot z\right) \cdot \left(y \cdot z\right)}{x \cdot \left(y + z\right) - y \cdot z}}} \]
      flip-+ [<=] 58.9%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right) + y \cdot z}} \]
      distribute-lft-in [=>] 58.9%	\[ 2 \cdot \sqrt{\color{blue}{\left(x \cdot y + x \cdot z\right)} + y \cdot z} \]
      add-sqr-sqrt [=>] 58.5%	\[ 2 \cdot \color{blue}{\left(\sqrt{\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}} \cdot \sqrt{\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}}\right)} \]
      pow2 [=>] 58.5%	\[ 2 \cdot \color{blue}{{\left(\sqrt{\sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}}\right)}^{2}} \]
      pow1/2 [=>] 58.5%	\[ 2 \cdot {\left(\sqrt{\color{blue}{{\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{0.5}}}\right)}^{2} \]
      sqrt-pow1 [=>] 58.5%	\[ 2 \cdot {\color{blue}{\left({\left(\left(x \cdot y + x \cdot z\right) + y \cdot z\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]
      associate-+l+ [=>] 58.5%	\[ 2 \cdot {\left({\color{blue}{\left(x \cdot y + \left(x \cdot z + y \cdot z\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      distribute-rgt-in [<=] 58.5%	\[ 2 \cdot {\left({\left(x \cdot y + \color{blue}{z \cdot \left(x + y\right)}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      fma-udef [<=] 58.7%	\[ 2 \cdot {\left({\color{blue}{\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \]
      metadata-eval [=>] 58.7%	\[ 2 \cdot {\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   4. Taylor expanded in z around 0 31.7%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(y \cdot x\right)}^{0.25}\right)}}^{2} \]

   5. Taylor expanded in y around -inf 48.0%

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right)}\right)}}^{2} \]

   if -7.2e53 < y < 6.5000000000000002e-257

   1. Initial program 87.4%

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

   2. Simplified 87.4%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}} \]
      Step-by-step derivation

      [Start] 87.4%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 87.4%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in y around 0 87.4%

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

   if 6.5000000000000002e-257 < y

   1. Initial program 64.3%

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

   2. Simplified 64.4%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}} \]
      Step-by-step derivation

      [Start] 64.3%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 64.4%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in z around inf 39.4%

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

   4. Applied egg-rr 44.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x + y} \cdot \sqrt{z}\right)} \]
      Step-by-step derivation

      [Start] 39.4%	\[ 2 \cdot \sqrt{\left(y + x\right) \cdot z} \]
      sqrt-prod [=>] 44.5%	\[ 2 \cdot \color{blue}{\left(\sqrt{y + x} \cdot \sqrt{z}\right)} \]
      +-commutative [=>] 44.5%	\[ 2 \cdot \left(\sqrt{\color{blue}{x + y}} \cdot \sqrt{z}\right) \]

   5. Simplified 44.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{y + x} \cdot \sqrt{z}\right)} \]
      Step-by-step derivation

      [Start] 44.5%	\[ 2 \cdot \left(\sqrt{x + y} \cdot \sqrt{z}\right) \]
      +-commutative [=>] 44.5%	\[ 2 \cdot \left(\sqrt{\color{blue}{y + x}} \cdot \sqrt{z}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.6%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2
Accuracy	94.8%
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3
Accuracy	84.4%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4
Accuracy	83.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5
Accuracy	71.0%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 6
Accuracy	70.3%
Cost	7104

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

Alternative 7
Accuracy	70.2%
Cost	7104

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

Alternative 8
Accuracy	69.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Accuracy	70.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 10
Accuracy	68.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 11
Accuracy	35.5%
Cost	6720

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

Reproduce

herbie shell --seed 2023167 
(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)))))