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

Average Accuracy: 69.0% → 97.8%

Time: 12.4s

Precision: binary64

Cost: 13572

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{\sqrt{-x}}{\sqrt{\frac{-1}{y + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-296)
   (* 2.0 (/ (sqrt (- x)) (sqrt (/ -1.0 (+ 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 <= 9.2e-296) {
		tmp = 2.0 * (sqrt(-x) / sqrt((-1.0 / (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 <= 9.2d-296) then
        tmp = 2.0d0 * (sqrt(-x) / sqrt(((-1.0d0) / (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 <= 9.2e-296) {
		tmp = 2.0 * (Math.sqrt(-x) / Math.sqrt((-1.0 / (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 <= 9.2e-296:
		tmp = 2.0 * (math.sqrt(-x) / math.sqrt((-1.0 / (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 <= 9.2e-296)
		tmp = Float64(2.0 * Float64(sqrt(Float64(-x)) / sqrt(Float64(-1.0 / 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 <= 9.2e-296)
		tmp = 2.0 * (sqrt(-x) / sqrt((-1.0 / (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, 9.2e-296], N[(2.0 * N[(N[Sqrt[(-x)], $MachinePrecision] / N[Sqrt[N[(-1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	69.0%
Target	82.2%
Herbie	97.8%

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

Derivation

1. Split input into 2 regimes

2. if y < 9.20000000000000016e-296

   1. Initial program 69.6%

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

   2. Simplified 69.6%

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

      [Start] 69.6	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.6	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 69.5%

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

   4. Applied egg-rr 47.8%

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

   5. Simplified 69.4%

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

      [Start] 47.8	\[ 2 \cdot \sqrt{\frac{x \cdot \left(y \cdot y - z \cdot z\right)}{y - z}} \]
      associate-/l* [=>] 58.0	\[ 2 \cdot \sqrt{\color{blue}{\frac{x}{\frac{y - z}{y \cdot y - z \cdot z}}}} \]
      difference-of-squares [=>] 58.0	\[ 2 \cdot \sqrt{\frac{x}{\frac{y - z}{\color{blue}{\left(y + z\right) \cdot \left(y - z\right)}}}} \]
      associate-/r* [=>] 69.4	\[ 2 \cdot \sqrt{\frac{x}{\color{blue}{\frac{\frac{y - z}{y + z}}{y - z}}}} \]
      +-commutative [=>] 69.4	\[ 2 \cdot \sqrt{\frac{x}{\frac{\frac{y - z}{\color{blue}{z + y}}}{y - z}}} \]

   6. Applied egg-rr 99.3%

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

   7. Simplified 99.3%

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

      [Start] 99.3	\[ 2 \cdot \frac{\sqrt{-x}}{\sqrt{-\frac{1}{y + z}}} \]
      distribute-neg-frac [=>] 99.3	\[ 2 \cdot \frac{\sqrt{-x}}{\sqrt{\color{blue}{\frac{-1}{y + z}}}} \]
      metadata-eval [=>] 99.3	\[ 2 \cdot \frac{\sqrt{-x}}{\sqrt{\frac{\color{blue}{-1}}{y + z}}} \]
      +-commutative [=>] 99.3	\[ 2 \cdot \frac{\sqrt{-x}}{\sqrt{\frac{-1}{\color{blue}{z + y}}}} \]

   if 9.20000000000000016e-296 < y

   1. Initial program 68.5%

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

   2. Simplified 68.5%

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

      [Start] 68.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 68.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 66.1%

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

   4. Applied egg-rr 96.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{\sqrt{-x}}{\sqrt{\frac{-1}{y + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.7%
Cost	14665

\[\begin{array}{l} t_0 := \left(y \cdot x + x \cdot z\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq \infty\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \end{array} \]

Alternative 2
Accuracy	69.0%
Cost	7104

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

Alternative 3
Accuracy	67.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Accuracy	68.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Accuracy	66.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 6
Accuracy	35.3%
Cost	6720

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

Reproduce

herbie shell --seed 2023129 
(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)))))