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

Average Error: 19.4 → 3.3

Time: 16.7s

Precision: binary64

Cost: 26828

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-230}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

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 (- (- z) y)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -1.05e+40)
     t_0
     (if (<= y -2.4e-230)
       (* 2.0 (sqrt (+ (* x z) (* y x))))
       (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((-z - y)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.05e+40) {
		tmp = t_0;
	} else if (y <= -2.4e-230) {
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	} 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((-z - y)) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-1.05d+40)) then
        tmp = t_0
    else if (y <= (-2.4d-230)) then
        tmp = 2.0d0 * sqrt(((x * z) + (y * x)))
    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((-z - y)) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -1.05e+40) {
		tmp = t_0;
	} else if (y <= -2.4e-230) {
		tmp = 2.0 * Math.sqrt(((x * z) + (y * x)));
	} 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((-z - y)) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -1.05e+40:
		tmp = t_0
	elif y <= -2.4e-230:
		tmp = 2.0 * math.sqrt(((x * z) + (y * x)))
	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(-z) - y)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -1.05e+40)
		tmp = t_0;
	elseif (y <= -2.4e-230)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * z) + Float64(y * x))));
	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((-z - y)) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -1.05e+40)
		tmp = t_0;
	elseif (y <= -2.4e-230)
		tmp = 2.0 * sqrt(((x * z) + (y * x)));
	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[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+40], t$95$0, If[LessEqual[y, -2.4e-230], N[(2.0 * N[Sqrt[N[(N[(x * z), $MachinePrecision] + N[(y * x), $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	19.4
Target	11.2
Herbie	3.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000005e40 or -2.4000000000000002e-230 < y < 0.0

   1. Initial program 40.2

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

   2. Taylor expanded in y around 0 40.2

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

   3. Applied egg-rr 40.2

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

   4. Applied egg-rr 40.4

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

   5. Taylor expanded in x around -inf 6.5

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

   if -1.05000000000000005e40 < y < -2.4000000000000002e-230

   1. Initial program 2.6

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

   2. Taylor expanded in x around inf 2.7

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

   3. Taylor expanded in y around 0 2.7

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

   if 0.0 < y

   1. Initial program 19.2

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

   2. Applied egg-rr 19.4

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

   3. Taylor expanded in x around 0 20.6

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

   4. Simplified 20.6

      \[\leadsto 2 \cdot \color{blue}{\sqrt{z \cdot y}} \]
      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 2.2

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-230}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.4
Cost	26828

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{\left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right) \cdot 0.16666666666666666}\right)}^{3}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-230}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	3.7
Cost	26564

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-x\right) - z\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	3.9
Cost	26436

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot {\left(e^{0.16666666666666666 \cdot \left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right)}\right)}^{3}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Error	11.2
Cost	13892

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot z + y \cdot x\right) + y \cdot z \leq 4 \cdot 10^{+299}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot \left(x + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 5
Error	19.4
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot z + y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	19.4
Cost	7104

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

Alternative 7
Error	20.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Error	19.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 9
Error	20.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 10
Error	61.8
Cost	6720

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

Alternative 11
Error	41.5
Cost	6720

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

Reproduce

herbie shell --seed 2022294 
(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)))))