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

Average Error: 19.6 → 3.1

Time: 15.0s

Precision: binary64

Cost: 26828

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -6.4 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -6.4e+28)
     t_0
     (if (<= y -2.9e-167)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y 0.0) t_0 (* 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 t_0 = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -6.4e+28) {
		tmp = t_0;
	} else if (y <= -2.9e-167) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= 0.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-6.4d+28)) then
        tmp = t_0
    else if (y <= (-2.9d-167)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= 0.0d0) then
        tmp = t_0
    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 t_0 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -6.4e+28) {
		tmp = t_0;
	} else if (y <= -2.9e-167) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= 0.0) {
		tmp = t_0;
	} 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):
	t_0 = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -6.4e+28:
		tmp = t_0
	elif y <= -2.9e-167:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= 0.0:
		tmp = t_0
	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)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -6.4e+28)
		tmp = t_0;
	elseif (y <= -2.9e-167)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= 0.0)
		tmp = t_0;
	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)
	t_0 = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -6.4e+28)
		tmp = t_0;
	elseif (y <= -2.9e-167)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= 0.0)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.4e+28], t$95$0, If[LessEqual[y, -2.9e-167], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0], t$95$0, N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	19.6
Target	11.5
Herbie	3.1

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

Derivation

1. Split input into 3 regimes

2. if y < -6.4000000000000001e28 or -2.90000000000000003e-167 < y < 0.0

   1. Initial program 34.3

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

   2. Applied egg-rr 34.4

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

   3. Taylor expanded in x around -inf 6.4

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

   if -6.4000000000000001e28 < y < -2.90000000000000003e-167

   1. Initial program 0.6

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

   2. Taylor expanded in x around inf 0.6

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

   if 0.0 < y

   1. Initial program 19.3

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

   2. Applied egg-rr 19.6

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

   3. Taylor expanded in x around 0 20.8

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

   4. Simplified 20.8

      \[\leadsto 2 \cdot \color{blue}{\sqrt{z \cdot y}} \]
      Proof
      (sqrt.f64 (*.f64 z y)): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 y z))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 2.4

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+28}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	14664

\[\begin{array}{l} t_0 := \left(x \cdot z + y \cdot x\right) + y \cdot z\\ t_1 := 2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-315}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.6
Cost	7104

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

Alternative 3
Error	20.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-269}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Error	19.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-269}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Error	21.1
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 6
Error	41.3
Cost	6720

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

Reproduce

herbie shell --seed 2022295 
(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)))))