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

Average Error: 19.9 → 2.8

Time: 12.9s

Precision: binary64

Cost: 13572

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-260}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(-x\right) - z}}{\sqrt{\frac{-1}{y}}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e-260)
   (* 2.0 (/ (sqrt (- (- x) z)) (sqrt (/ -1.0 y))))
   (if (<= y 6e+62)
     (* 2.0 (sqrt (+ (* x (+ y z)) (* y z))))
     (* 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 tmp;
	if (y <= -3.2e-260) {
		tmp = 2.0 * (sqrt((-x - z)) / sqrt((-1.0 / y)));
	} else if (y <= 6e+62) {
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	} 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) :: tmp
    if (y <= (-3.2d-260)) then
        tmp = 2.0d0 * (sqrt((-x - z)) / sqrt(((-1.0d0) / y)))
    else if (y <= 6d+62) then
        tmp = 2.0d0 * sqrt(((x * (y + z)) + (y * z)))
    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 tmp;
	if (y <= -3.2e-260) {
		tmp = 2.0 * (Math.sqrt((-x - z)) / Math.sqrt((-1.0 / y)));
	} else if (y <= 6e+62) {
		tmp = 2.0 * Math.sqrt(((x * (y + z)) + (y * z)));
	} 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):
	tmp = 0
	if y <= -3.2e-260:
		tmp = 2.0 * (math.sqrt((-x - z)) / math.sqrt((-1.0 / y)))
	elif y <= 6e+62:
		tmp = 2.0 * math.sqrt(((x * (y + z)) + (y * z)))
	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)
	tmp = 0.0
	if (y <= -3.2e-260)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(-x) - z)) / sqrt(Float64(-1.0 / y))));
	elseif (y <= 6e+62)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * Float64(y + z)) + Float64(y * z))));
	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)
	tmp = 0.0;
	if (y <= -3.2e-260)
		tmp = 2.0 * (sqrt((-x - z)) / sqrt((-1.0 / y)));
	elseif (y <= 6e+62)
		tmp = 2.0 * sqrt(((x * (y + z)) + (y * z)));
	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_] := If[LessEqual[y, -3.2e-260], N[(2.0 * N[(N[Sqrt[N[((-x) - z), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+62], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.9
Target	11.5
Herbie	2.8

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

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999995e-260

   1. Initial program 19.7

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

   2. Simplified 19.7

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

      [Start] 19.7	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 19.7	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 19.9

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}}} \]

   4. Taylor expanded in y around inf 21.3

      \[\leadsto 2 \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{y \cdot \left(z + x\right)}}}} \]

   5. Simplified 21.3

      \[\leadsto 2 \cdot \sqrt{\frac{1}{\color{blue}{\frac{\frac{1}{y}}{z + x}}}} \]
      Proof

      [Start] 21.3	\[ 2 \cdot \sqrt{\frac{1}{\frac{1}{y \cdot \left(z + x\right)}}} \]
      associate-/r* [=>] 21.3	\[ 2 \cdot \sqrt{\frac{1}{\color{blue}{\frac{\frac{1}{y}}{z + x}}}} \]

   6. Applied egg-rr 2.1

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

   7. Simplified 2.0

      \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{\left(-x\right) - z}}{\sqrt{\frac{-1}{y}}}} \]
      Proof

      [Start] 2.1	\[ 2 \cdot \left(\frac{1}{\sqrt{\frac{-1}{y}}} \cdot \sqrt{-\left(z + x\right)}\right) \]
      associate-*l/ [=>] 2.0	\[ 2 \cdot \color{blue}{\frac{1 \cdot \sqrt{-\left(z + x\right)}}{\sqrt{\frac{-1}{y}}}} \]
      *-lft-identity [=>] 2.0	\[ 2 \cdot \frac{\color{blue}{\sqrt{-\left(z + x\right)}}}{\sqrt{\frac{-1}{y}}} \]
      +-commutative [=>] 2.0	\[ 2 \cdot \frac{\sqrt{-\color{blue}{\left(x + z\right)}}}{\sqrt{\frac{-1}{y}}} \]
      distribute-neg-in [=>] 2.0	\[ 2 \cdot \frac{\sqrt{\color{blue}{\left(-x\right) + \left(-z\right)}}}{\sqrt{\frac{-1}{y}}} \]
      unsub-neg [=>] 2.0	\[ 2 \cdot \frac{\sqrt{\color{blue}{\left(-x\right) - z}}}{\sqrt{\frac{-1}{y}}} \]

   if -3.19999999999999995e-260 < y < 6e62

   1. Initial program 5.0

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

   2. Simplified 5.0

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

      [Start] 5.0	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 5.0	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   if 6e62 < y

   1. Initial program 48.3

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

   2. Simplified 48.3

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

      [Start] 48.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 48.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 48.5

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

   4. Applied egg-rr 1.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-260}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(-x\right) - z}}{\sqrt{\frac{-1}{y}}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+62}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	26820

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(-x\right) - z}}{\sqrt{\frac{-1}{y}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y} + 0.5 \cdot \left(\frac{x}{\sqrt{y}} \cdot \frac{y + z}{\sqrt{z}}\right)\right)\\ \end{array} \]

Alternative 2
Error	11.2
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \cdot z + \left(y \cdot x + x \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	19.4
Cost	7876

\[\begin{array}{l} \mathbf{if}\;y \cdot z + \left(y \cdot x + x \cdot z\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y}}{z}\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Error	19.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-284}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Error	20.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	20.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-292}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	21.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Error	41.7
Cost	6720

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

Reproduce

herbie shell --seed 2022354 
(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)))))