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

Average Error: 20.0 → 2.9

Time: 22.3s

Precision: binary64

Cost: 26436

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05e+27)
   (* 2.0 (pow (exp (* 0.25 (- (log (- x)) (log (/ -1.0 y))))) 2.0))
   (if (<= y 0.0)
     (* 2.0 (sqrt (* x (+ y z))))
     (* 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 tmp;
	if (y <= -1.05e+27) {
		tmp = 2.0 * pow(exp((0.25 * (log(-x) - log((-1.0 / y))))), 2.0);
	} else if (y <= 0.0) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} 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) :: tmp
    if (y <= (-1.05d+27)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log(-x) - log(((-1.0d0) / y))))) ** 2.0d0)
    else if (y <= 0.0d0) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    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 tmp;
	if (y <= -1.05e+27) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log(-x) - Math.log((-1.0 / y))))), 2.0);
	} else if (y <= 0.0) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} 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):
	tmp = 0
	if y <= -1.05e+27:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log(-x) - math.log((-1.0 / y))))), 2.0)
	elif y <= 0.0:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	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)
	tmp = 0.0
	if (y <= -1.05e+27)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(-x)) - log(Float64(-1.0 / y))))) ^ 2.0));
	elseif (y <= 0.0)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	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)
	tmp = 0.0;
	if (y <= -1.05e+27)
		tmp = 2.0 * (exp((0.25 * (log(-x) - log((-1.0 / y))))) ^ 2.0);
	elseif (y <= 0.0)
		tmp = 2.0 * sqrt((x * (y + z)));
	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_] := If[LessEqual[y, -1.05e+27], N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[(-x)], $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	20.0
Target	11.9
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999997e27

   1. Initial program 41.8

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

   2. Applied egg-rr 41.9

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.25}\right)}^{2}} \]

   3. Taylor expanded in z around 0 42.3

      \[\leadsto 2 \cdot {\left({\color{blue}{\left(y \cdot x\right)}}^{0.25}\right)}^{2} \]

   4. Taylor expanded in y around -inf 7.2

      \[\leadsto 2 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right)}\right)}^{2}} \]

   if -1.04999999999999997e27 < y < 0.0

   1. Initial program 3.9

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

   2. Taylor expanded in x around inf 3.9

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

   if 0.0 < y

   1. Initial program 19.6

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

   2. Simplified 19.6

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]
      Proof
      (*.f64 2 (sqrt.f64 (fma.f64 x y (*.f64 z (+.f64 x y))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (fma.f64 x y (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x z) (*.f64 y z)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 19.6

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

   4. Applied egg-rr 39.5

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

   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 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.5
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 2
Error	3.2
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3
Error	10.7
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(y, x + z, x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4
Error	10.7
Cost	13380

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

Alternative 5
Error	12.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-194}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right) + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 6
Error	20.0
Cost	7104

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

Alternative 7
Error	21.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Error	20.1
Cost	6980

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

Alternative 9
Error	21.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 10
Error	42.2
Cost	6720

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

Reproduce

herbie shell --seed 2022290 
(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)))))