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

Average Error: 19.5 → 2.1

Time: 11.8s

Precision: binary64

Cost: 20364

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-195}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot x}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

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 (- (- z) y)) (log (/ -1.0 x))) 0.5)))))
   (if (<= y -3e+43)
     t_0
     (if (<= y -5.2e-195)
       (* 2.0 (sqrt (+ (* z x) (* y x))))
       (if (<= y -2.6e-306) t_0 (* 2.0 (* (sqrt (+ y x)) (sqrt z))))))))

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((-z - y)) - log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -3e+43) {
		tmp = t_0;
	} else if (y <= -5.2e-195) {
		tmp = 2.0 * sqrt(((z * x) + (y * x)));
	} else if (y <= -2.6e-306) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	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((-z - y)) - log(((-1.0d0) / x))) * 0.5d0))
    if (y <= (-3d+43)) then
        tmp = t_0
    else if (y <= (-5.2d-195)) then
        tmp = 2.0d0 * sqrt(((z * x) + (y * x)))
    else if (y <= (-2.6d-306)) then
        tmp = t_0
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    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((-z - y)) - Math.log((-1.0 / x))) * 0.5));
	double tmp;
	if (y <= -3e+43) {
		tmp = t_0;
	} else if (y <= -5.2e-195) {
		tmp = 2.0 * Math.sqrt(((z * x) + (y * x)));
	} else if (y <= -2.6e-306) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	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((-z - y)) - math.log((-1.0 / x))) * 0.5))
	tmp = 0
	if y <= -3e+43:
		tmp = t_0
	elif y <= -5.2e-195:
		tmp = 2.0 * math.sqrt(((z * x) + (y * x)))
	elif y <= -2.6e-306:
		tmp = t_0
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	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(-z) - y)) - log(Float64(-1.0 / x))) * 0.5)))
	tmp = 0.0
	if (y <= -3e+43)
		tmp = t_0;
	elseif (y <= -5.2e-195)
		tmp = Float64(2.0 * sqrt(Float64(Float64(z * x) + Float64(y * x))));
	elseif (y <= -2.6e-306)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	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((-z - y)) - log((-1.0 / x))) * 0.5));
	tmp = 0.0;
	if (y <= -3e+43)
		tmp = t_0;
	elseif (y <= -5.2e-195)
		tmp = 2.0 * sqrt(((z * x) + (y * x)));
	elseif (y <= -2.6e-306)
		tmp = t_0;
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	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[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+43], t$95$0, If[LessEqual[y, -5.2e-195], N[(2.0 * N[Sqrt[N[(N[(z * x), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.6e-306], t$95$0, N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	19.5
Target	11.1
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if y < -3.00000000000000017e43 or -5.2000000000000003e-195 < y < -2.6e-306

   1. Initial program 38.8

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

   2. Simplified 38.8

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

      [Start] 38.8	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 38.8	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 40.8

      \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in x around -inf 6.6

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

   5. Simplified 6.6

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

      [Start] 6.6	\[ 2 \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot \left(y + z\right)\right)\right) \cdot 0.5} \]
      +-commutative [=>] 6.6	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) + -1 \cdot \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 6.6	\[ 2 \cdot e^{\left(\log \left(-1 \cdot \left(y + z\right)\right) + \color{blue}{\left(-\log \left(\frac{-1}{x}\right)\right)}\right) \cdot 0.5} \]
      unsub-neg [=>] 6.6	\[ 2 \cdot e^{\color{blue}{\left(\log \left(-1 \cdot \left(y + z\right)\right) - \log \left(\frac{-1}{x}\right)\right)} \cdot 0.5} \]
      mul-1-neg [=>] 6.6	\[ 2 \cdot e^{\left(\log \color{blue}{\left(-\left(y + z\right)\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      neg-sub0 [=>] 6.6	\[ 2 \cdot e^{\left(\log \color{blue}{\left(0 - \left(y + z\right)\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      +-commutative [=>] 6.6	\[ 2 \cdot e^{\left(\log \left(0 - \color{blue}{\left(z + y\right)}\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      associate--r+ [=>] 6.6	\[ 2 \cdot e^{\left(\log \color{blue}{\left(\left(0 - z\right) - y\right)} - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]
      neg-sub0 [<=] 6.6	\[ 2 \cdot e^{\left(\log \left(\color{blue}{\left(-z\right)} - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5} \]

   if -3.00000000000000017e43 < y < -5.2000000000000003e-195

   1. Initial program 1.3

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

   2. Simplified 1.3

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

      [Start] 1.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 1.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 1.3

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

   4. Taylor expanded in y around 0 1.3

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

   if -2.6e-306 < y

   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 22.9

      \[\leadsto 2 \cdot \color{blue}{e^{\log \left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right) \cdot 0.5}} \]

   4. Taylor expanded in z around inf 22.9

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

   5. Applied egg-rr 0.4

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+43}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-195}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot x}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-306}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.9
Cost	13892

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot x + y \cdot x\right) + y \cdot z \leq 10^{+305}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot \left(z + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	10.0
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3
Error	19.6
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot x + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Error	19.5
Cost	7104

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

Alternative 5
Error	19.5
Cost	7104

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

Alternative 6
Error	20.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	19.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-306}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Error	20.9
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 9
Error	41.4
Cost	6720

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

Reproduce

herbie shell --seed 2023066 
(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)))))