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

Average Error: 20.2 → 2.9

Time: 13.6s

Precision: binary64

Cost: 26564

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y + z\right) \cdot \left(y - z\right)}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{\frac{x}{\frac{1}{y + z}} + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+154)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- z) y)) (log (/ -1.0 x))))) 2.0))
   (if (<= y -6.6e+40)
     (* 2.0 (/ (sqrt (* (+ y z) (- y z))) (sqrt (/ (- y z) x))))
     (if (<= y 1.2e-280)
       (* 2.0 (sqrt (+ (/ x (/ 1.0 (+ y z))) (* y z))))
       (* 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 <= -1.35e+154) {
		tmp = 2.0 * pow(exp((0.25 * (log((-z - y)) - log((-1.0 / x))))), 2.0);
	} else if (y <= -6.6e+40) {
		tmp = 2.0 * (sqrt(((y + z) * (y - z))) / sqrt(((y - z) / x)));
	} else if (y <= 1.2e-280) {
		tmp = 2.0 * sqrt(((x / (1.0 / (y + z))) + (y * z)));
	} 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 <= (-1.35d+154)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-z - y)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else if (y <= (-6.6d+40)) then
        tmp = 2.0d0 * (sqrt(((y + z) * (y - z))) / sqrt(((y - z) / x)))
    else if (y <= 1.2d-280) then
        tmp = 2.0d0 * sqrt(((x / (1.0d0 / (y + z))) + (y * z)))
    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 <= -1.35e+154) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-z - y)) - Math.log((-1.0 / x))))), 2.0);
	} else if (y <= -6.6e+40) {
		tmp = 2.0 * (Math.sqrt(((y + z) * (y - z))) / Math.sqrt(((y - z) / x)));
	} else if (y <= 1.2e-280) {
		tmp = 2.0 * Math.sqrt(((x / (1.0 / (y + z))) + (y * z)));
	} 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 <= -1.35e+154:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-z - y)) - math.log((-1.0 / x))))), 2.0)
	elif y <= -6.6e+40:
		tmp = 2.0 * (math.sqrt(((y + z) * (y - z))) / math.sqrt(((y - z) / x)))
	elif y <= 1.2e-280:
		tmp = 2.0 * math.sqrt(((x / (1.0 / (y + z))) + (y * z)))
	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 <= -1.35e+154)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= -6.6e+40)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(y + z) * Float64(y - z))) / sqrt(Float64(Float64(y - z) / x))));
	elseif (y <= 1.2e-280)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x / Float64(1.0 / Float64(y + z))) + Float64(y * z))));
	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 <= -1.35e+154)
		tmp = 2.0 * (exp((0.25 * (log((-z - y)) - log((-1.0 / x))))) ^ 2.0);
	elseif (y <= -6.6e+40)
		tmp = 2.0 * (sqrt(((y + z) * (y - z))) / sqrt(((y - z) / x)));
	elseif (y <= 1.2e-280)
		tmp = 2.0 * sqrt(((x / (1.0 / (y + z))) + (y * z)));
	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, -1.35e+154], 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, -6.6e+40], N[(2.0 * N[(N[Sqrt[N[(N[(y + z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(y - z), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-280], N[(2.0 * N[Sqrt[N[(N[(x / N[(1.0 / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	20.2
Target	11.6
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 4 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 64.0	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 64.0

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

   4. Taylor expanded in x around -inf 7.1

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

   if -1.35000000000000003e154 < y < -6.5999999999999997e40

   1. Initial program 30.5

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

   2. Simplified 30.5

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

      [Start] 30.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 30.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around inf 30.5

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

   4. Applied egg-rr 49.7

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

   5. Applied egg-rr 0.3

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

   6. Simplified 0.3

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

      [Start] 0.3	\[ 2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \left(y + z\right)}}{\sqrt{\frac{y - z}{x}}} \]
      +-commutative [=>] 0.3	\[ 2 \cdot \frac{\sqrt{\left(y - z\right) \cdot \color{blue}{\left(z + y\right)}}}{\sqrt{\frac{y - z}{x}}} \]

   if -6.5999999999999997e40 < y < 1.1999999999999999e-280

   1. Initial program 4.2

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

   2. Simplified 4.2

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

      [Start] 4.2	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 4.2	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 4.3

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

   if 1.1999999999999999e-280 < y

   1. Initial program 20.0

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

   2. Simplified 20.0

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

      [Start] 20.0	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 20.0	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 21.3

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

   4. Applied egg-rr 2.1

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{y}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y + z\right) \cdot \left(y - z\right)}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{\frac{x}{\frac{1}{y + z}} + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(y + z\right) \cdot \left(y - z\right)}}{\sqrt{\frac{y - z}{x}}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	11.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	11.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Error	20.2
Cost	7104

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

Alternative 5
Error	20.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	20.3
Cost	6980

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

Alternative 7
Error	21.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Error	42.1
Cost	6720

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

Reproduce

herbie shell --seed 2023016 
(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)))))