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

Average Error: 30.27% → 11.04%

Time: 13.1s

Precision: binary64

Cost: 13892

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \frac{\sqrt{y \cdot y - z \cdot z}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-255}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+39)
   (* 2.0 (/ (sqrt (- (* y y) (* z z))) (sqrt (/ (- y z) x))))
   (if (<= y 3.5e-255)
     (* 2.0 (sqrt (+ (* z (+ y x)) (* y x))))
     (* 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 <= -5.8e+39) {
		tmp = 2.0 * (sqrt(((y * y) - (z * z))) / sqrt(((y - z) / x)));
	} else if (y <= 3.5e-255) {
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	} 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 <= (-5.8d+39)) then
        tmp = 2.0d0 * (sqrt(((y * y) - (z * z))) / sqrt(((y - z) / x)))
    else if (y <= 3.5d-255) then
        tmp = 2.0d0 * sqrt(((z * (y + x)) + (y * x)))
    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 <= -5.8e+39) {
		tmp = 2.0 * (Math.sqrt(((y * y) - (z * z))) / Math.sqrt(((y - z) / x)));
	} else if (y <= 3.5e-255) {
		tmp = 2.0 * Math.sqrt(((z * (y + x)) + (y * x)));
	} 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 <= -5.8e+39:
		tmp = 2.0 * (math.sqrt(((y * y) - (z * z))) / math.sqrt(((y - z) / x)))
	elif y <= 3.5e-255:
		tmp = 2.0 * math.sqrt(((z * (y + x)) + (y * x)))
	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 <= -5.8e+39)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(y * y) - Float64(z * z))) / sqrt(Float64(Float64(y - z) / x))));
	elseif (y <= 3.5e-255)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z * Float64(y + x)) + Float64(y * x))));
	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 <= -5.8e+39)
		tmp = 2.0 * (sqrt(((y * y) - (z * z))) / sqrt(((y - z) / x)));
	elseif (y <= 3.5e-255)
		tmp = 2.0 * sqrt(((z * (y + x)) + (y * x)));
	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, -5.8e+39], N[(2.0 * N[(N[Sqrt[N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(y - z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-255], N[(2.0 * N[Sqrt[N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	30.27%
Target	16.99%
Herbie	11.04%

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

Derivation

1. Split input into 3 regimes

2. if y < -5.80000000000000059e39

   1. Initial program 69.44

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

   2. Simplified 69.43

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

      [Start] 69.44	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 69.43	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 69.45

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

   4. Applied egg-rr 87.1

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

   5. Applied egg-rr 39.72

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

   6. Simplified 39.72

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

      [Start] 39.72	\[ 2 \cdot \frac{\sqrt{\mathsf{fma}\left(y, y, z \cdot \left(-z\right)\right)}}{\sqrt{\frac{y - z}{x}}} \]
      distribute-rgt-neg-out [=>] 39.72	\[ 2 \cdot \frac{\sqrt{\mathsf{fma}\left(y, y, \color{blue}{-z \cdot z}\right)}}{\sqrt{\frac{y - z}{x}}} \]
      fma-neg [<=] 39.72	\[ 2 \cdot \frac{\sqrt{\color{blue}{y \cdot y - z \cdot z}}}{\sqrt{\frac{y - z}{x}}} \]

   if -5.80000000000000059e39 < y < 3.49999999999999979e-255

   1. Initial program 6.5

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

   2. Simplified 6.48

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

      [Start] 6.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      associate-+l+ [=>] 6.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]
      fma-def [=>] 6.49	\[ 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(x, y, x \cdot z + y \cdot z\right)}} \]
      distribute-rgt-out [=>] 6.48	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

   3. Applied egg-rr 6.49

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

   if 3.49999999999999979e-255 < y

   1. Initial program 30.23

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

   2. Simplified 30.23

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

      [Start] 30.23	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 30.23	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 32.21

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

   4. Applied egg-rr 3.03

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

4. Final simplification 11.04

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \frac{\sqrt{y \cdot y - z \cdot z}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-255}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.07%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-282}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{x}}{y + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	30.28%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 3
Error	30.26%
Cost	7104

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

Alternative 4
Error	30.26%
Cost	7104

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

Alternative 5
Error	31.73%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-264}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	30.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-299}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	32.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Error	65.57%
Cost	6720

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

Reproduce

herbie shell --seed 2023089 
(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)))))