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

Percentage Accurate: 70.2% → 94.7%

Time: 16.3s

Precision: binary64

Cost: 20364

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

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-221}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (* (- (log (- (- y) z)) (log (/ -1.0 x))) 0.5)))))
   (if (<= y -5.6e+27)
     t_0
     (if (<= y -6e-221)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 6.2e-271) t_0 (* 2.0 (* (sqrt z) (sqrt y))))))))

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 t_0 = 2.0 * exp(((log((-y - z)) - log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -5.6e+27) {
		tmp = t_0;
	} else if (y <= -6e-221) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 6.2e-271) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp(((log((-y - z)) - log(((-1.0d0) / x))) * 0.5d0))
    if (y <= (-5.6d+27)) then
        tmp = t_0
    else if (y <= (-6d-221)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 6.2d-271) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt(z) * sqrt(y))
    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 t_0 = 2.0 * Math.exp(((Math.log((-y - z)) - Math.log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -5.6e+27) {
		tmp = t_0;
	} else if (y <= -6e-221) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 6.2e-271) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt(z) * Math.sqrt(y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = 2.0 * math.exp(((math.log((-y - z)) - math.log((-1.0 / x))) * 0.5))
	tmp = 0
	if y <= -5.6e+27:
		tmp = t_0
	elif y <= -6e-221:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 6.2e-271:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt(z) * math.sqrt(y))
	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)
	t_0 = Float64(2.0 * exp(Float64(Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))) * 0.5)))
	tmp = 0.0
	if (y <= -5.6e+27)
		tmp = t_0;
	elseif (y <= -6e-221)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 6.2e-271)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(z) * sqrt(y)));
	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)
	t_0 = 2.0 * exp(((log((-y - z)) - log((-1.0 / x))) * 0.5));
	tmp = 0.0;
	if (y <= -5.6e+27)
		tmp = t_0;
	elseif (y <= -6e-221)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 6.2e-271)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt(z) * sqrt(y));
	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_] := Block[{t$95$0 = N[(2.0 * N[Exp[N[(N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6e+27], t$95$0, If[LessEqual[y, -6e-221], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-271], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	70.2%
Target	82.7%
Herbie	94.7%

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

Derivation

1. Split input into 3 regimes

2. if y < -5.5999999999999999e27 or -6.0000000000000003e-221 < y < 6.1999999999999998e-271

   1. Initial program 65.2%

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

   2. Simplified 65.3%

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

      [Start] 65.2%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 65.3%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 48.5%

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

   4. Applied egg-rr 45.0%

      \[\leadsto 2 \cdot \color{blue}{e^{\log \left(x \cdot \left(y + z\right)\right) \cdot 0.5}} \]
      Step-by-step derivation

      [Start] 48.5%	\[ 2 \cdot \sqrt{\left(y + z\right) \cdot x} \]
      pow1/2 [=>] 48.6%	\[ 2 \cdot \color{blue}{{\left(\left(y + z\right) \cdot x\right)}^{0.5}} \]
      pow-to-exp [=>] 45.0%	\[ 2 \cdot \color{blue}{e^{\log \left(\left(y + z\right) \cdot x\right) \cdot 0.5}} \]
      *-commutative [=>] 45.0%	\[ 2 \cdot e^{\log \color{blue}{\left(x \cdot \left(y + z\right)\right)} \cdot 0.5} \]

   5. Taylor expanded in x around -inf 44.2%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right)} \cdot 0.5} \]

   6. Simplified 44.2%

      \[\leadsto 2 \cdot e^{\color{blue}{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      Step-by-step derivation

      [Start] 44.2%	\[ 2 \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right) \cdot 0.5} \]
      +-commutative [=>] 44.2%	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 44.2%	\[ 2 \cdot e^{\left(\log \left(-1 \cdot \left(y + z\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{x}\right)\right)}\right) \cdot 0.5} \]
      unsub-neg [=>] 44.2%	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      distribute-lft-in [=>] 44.2%	\[ 2 \cdot e^{\left(\log \color{blue}{\left(-1 \cdot y + -1 \cdot z\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      neg-mul-1 [<=] 44.2%	\[ 2 \cdot e^{\left(\log \left(-1 \cdot y + \color{blue}{\left(-z\right)}\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      unsub-neg [=>] 44.2%	\[ 2 \cdot e^{\left(\log \color{blue}{\left(-1 \cdot y - z\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      mul-1-neg [=>] 44.2%	\[ 2 \cdot e^{\left(\log \left(\color{blue}{\left(-y\right)} - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]

   if -5.5999999999999999e27 < y < -6.0000000000000003e-221

   1. Initial program 83.8%

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

   2. Simplified 83.8%

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

      [Start] 83.8%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 83.8%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 47.2%

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

   if 6.1999999999999998e-271 < y

   1. Initial program 70.3%

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

   2. Simplified 70.3%

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

      [Start] 70.3%	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 70.3%	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 28.0%

      \[\leadsto 2 \cdot \color{blue}{\sqrt{y \cdot z}} \]

   4. Applied egg-rr 35.7%

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

      [Start] 28.0%	\[ 2 \cdot \sqrt{y \cdot z} \]
      *-commutative [=>] 28.0%	\[ 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]
      sqrt-prod [=>] 35.7%	\[ 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-221}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.7%
Cost	20364

\[\begin{array}{l} t_0 := 2 \cdot e^{\left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-221}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	94.2%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot e^{0.5 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Accuracy	89.3%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \left(\sqrt{y \cdot y - z \cdot z} \cdot \sqrt{\frac{x}{y - z}}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Accuracy	89.2%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{\sqrt{y \cdot y - z \cdot z}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5
Accuracy	83.1%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(z, y + x, y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 6
Accuracy	83.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 7
Accuracy	70.3%
Cost	7104

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

Alternative 8
Accuracy	69.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Accuracy	70.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 10
Accuracy	68.0%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(y \cdot z\right)}^{0.5}\\ \end{array} \]

Alternative 11
Accuracy	68.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 12
Accuracy	35.2%
Cost	6720

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

Reproduce

herbie shell --seed 2023271 
(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)))))