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

Average Error: 31.93% → 3.51%

Time: 16.1s

Precision: binary64

Cost: 13572

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{\sqrt{-y}}{\sqrt{\frac{-1}{z + x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (* 2.0 (/ (sqrt (- y)) (sqrt (/ -1.0 (+ z 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 <= -5e-310) {
		tmp = 2.0 * (sqrt(-y) / sqrt((-1.0 / (z + 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 <= (-5d-310)) then
        tmp = 2.0d0 * (sqrt(-y) / sqrt(((-1.0d0) / (z + 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 <= -5e-310) {
		tmp = 2.0 * (Math.sqrt(-y) / Math.sqrt((-1.0 / (z + 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 <= -5e-310:
		tmp = 2.0 * (math.sqrt(-y) / math.sqrt((-1.0 / (z + 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 <= -5e-310)
		tmp = Float64(2.0 * Float64(sqrt(Float64(-y)) / sqrt(Float64(-1.0 / Float64(z + 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 <= -5e-310)
		tmp = 2.0 * (sqrt(-y) / sqrt((-1.0 / (z + 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, -5e-310], N[(2.0 * N[(N[Sqrt[(-y)], $MachinePrecision] / N[Sqrt[N[(-1.0 / N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	31.93%
Target	18.6%
Herbie	3.51%

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 31.84

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

   2. Simplified 31.83

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

      [Start] 31.84	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      associate-+l+ [=>] 31.84	\[ 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]
      fma-def [=>] 31.83	\[ 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(x, y, x \cdot z + y \cdot z\right)}} \]
      distribute-rgt-out [=>] 31.83	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

   3. Applied egg-rr 31.83

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

   4. Applied egg-rr 63.47

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

   5. Simplified 74.93

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

      [Start] 63.47	\[ 2 \cdot \sqrt{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - {\left(z \cdot \left(x + y\right)\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      associate-*r* [=>] 68.26	\[ 2 \cdot \sqrt{\frac{\color{blue}{\left(\left(x \cdot y\right) \cdot x\right) \cdot y} - {\left(z \cdot \left(x + y\right)\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      *-commutative [=>] 68.26	\[ 2 \cdot \sqrt{\frac{\color{blue}{y \cdot \left(\left(x \cdot y\right) \cdot x\right)} - {\left(z \cdot \left(x + y\right)\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      *-commutative [<=] 68.26	\[ 2 \cdot \sqrt{\frac{y \cdot \left(\color{blue}{\left(y \cdot x\right)} \cdot x\right) - {\left(z \cdot \left(x + y\right)\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      associate-*l* [=>] 74.93	\[ 2 \cdot \sqrt{\frac{y \cdot \color{blue}{\left(y \cdot \left(x \cdot x\right)\right)} - {\left(z \cdot \left(x + y\right)\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      +-commutative [=>] 74.93	\[ 2 \cdot \sqrt{\frac{y \cdot \left(y \cdot \left(x \cdot x\right)\right) - {\left(z \cdot \color{blue}{\left(y + x\right)}\right)}^{2}}{x \cdot y - z \cdot \left(x + y\right)}} \]
      *-commutative [<=] 74.93	\[ 2 \cdot \sqrt{\frac{y \cdot \left(y \cdot \left(x \cdot x\right)\right) - {\left(z \cdot \left(y + x\right)\right)}^{2}}{\color{blue}{y \cdot x} - z \cdot \left(x + y\right)}} \]
      +-commutative [=>] 74.93	\[ 2 \cdot \sqrt{\frac{y \cdot \left(y \cdot \left(x \cdot x\right)\right) - {\left(z \cdot \left(y + x\right)\right)}^{2}}{y \cdot x - z \cdot \color{blue}{\left(y + x\right)}}} \]

   6. Taylor expanded in y around inf 69.53

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

   7. Simplified 63.36

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

      [Start] 69.53	\[ 2 \cdot \sqrt{\frac{y \cdot \left({x}^{2} - {z}^{2}\right)}{x - z}} \]
      associate-/l* [=>] 63.36	\[ 2 \cdot \sqrt{\color{blue}{\frac{y}{\frac{x - z}{{x}^{2} - {z}^{2}}}}} \]
      unpow2 [=>] 63.36	\[ 2 \cdot \sqrt{\frac{y}{\frac{x - z}{\color{blue}{x \cdot x} - {z}^{2}}}} \]
      unpow2 [=>] 63.36	\[ 2 \cdot \sqrt{\frac{y}{\frac{x - z}{x \cdot x - \color{blue}{z \cdot z}}}} \]

   8. Applied egg-rr 3.62

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

   9. Simplified 3.62

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

      [Start] 3.62	\[ 2 \cdot \frac{\sqrt{-y}}{\sqrt{\frac{-1}{x + z}}} \]
      +-commutative [=>] 3.62	\[ 2 \cdot \frac{\sqrt{-y}}{\sqrt{\frac{-1}{\color{blue}{z + x}}}} \]

   if -4.999999999999985e-310 < y

   1. Initial program 32.03

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

   2. Simplified 32.04

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

      [Start] 32.03	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      associate-+l+ [=>] 32.03	\[ 2 \cdot \sqrt{\color{blue}{x \cdot y + \left(x \cdot z + y \cdot z\right)}} \]
      fma-def [=>] 32.03	\[ 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(x, y, x \cdot z + y \cdot z\right)}} \]
      distribute-rgt-out [=>] 32.04	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x + y\right)}\right)} \]

   3. Applied egg-rr 76.68

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

   4. Simplified 74.35

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

      [Start] 76.68	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \frac{\left(z \cdot z\right) \cdot \left(x \cdot x\right) - \left(y \cdot y\right) \cdot \left(z \cdot z\right)}{z \cdot \left(x - y\right)}\right)} \]
      unswap-sqr [=>] 76.72	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \frac{\color{blue}{\left(z \cdot x\right) \cdot \left(z \cdot x\right)} - \left(y \cdot y\right) \cdot \left(z \cdot z\right)}{z \cdot \left(x - y\right)}\right)} \]
      associate-*l* [=>] 74.35	\[ 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \frac{\left(z \cdot x\right) \cdot \left(z \cdot x\right) - \color{blue}{y \cdot \left(y \cdot \left(z \cdot z\right)\right)}}{z \cdot \left(x - y\right)}\right)} \]

   5. Taylor expanded in x around 0 34.11

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

   6. Simplified 34.11

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

      [Start] 34.11	\[ 2 \cdot \sqrt{y \cdot z} \]
      *-commutative [<=] 34.11	\[ 2 \cdot \sqrt{\color{blue}{z \cdot y}} \]

   7. Applied egg-rr 3.4

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

4. Final simplification 3.51

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{\sqrt{-y}}{\sqrt{\frac{-1}{z + x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.75%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	17.87%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	30.54%
Cost	7876

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

Alternative 4
Error	32.05%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Error	31.93%
Cost	7104

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

Alternative 6
Error	31.93%
Cost	7104

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

Alternative 7
Error	33.23%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-273}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Error	32.05%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-289}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Error	34.05%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(y \cdot z\right)}^{0.5}\\ \end{array} \]

Alternative 10
Error	34.05%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 11
Error	65.31%
Cost	6720

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

Reproduce

herbie shell --seed 2023121 
(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)))))