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

Average Error: 19.5 → 3.8

Time: 14.3s

Precision: binary64

Cost: 20228

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \frac{1}{e^{-0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.6e+36)
   (* 2.0 (/ 1.0 (exp (* -0.5 (- (log (- (- y) z)) (log (/ -1.0 x)))))))
   (if (<= y 4.7e+27)
     (* 2.0 (sqrt (+ (* x (+ 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 <= -5.6e+36) {
		tmp = 2.0 * (1.0 / exp((-0.5 * (log((-y - z)) - log((-1.0 / x))))));
	} else if (y <= 4.7e+27) {
		tmp = 2.0 * sqrt(((x * (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 <= (-5.6d+36)) then
        tmp = 2.0d0 * (1.0d0 / exp(((-0.5d0) * (log((-y - z)) - log(((-1.0d0) / x))))))
    else if (y <= 4.7d+27) then
        tmp = 2.0d0 * sqrt(((x * (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 <= -5.6e+36) {
		tmp = 2.0 * (1.0 / Math.exp((-0.5 * (Math.log((-y - z)) - Math.log((-1.0 / x))))));
	} else if (y <= 4.7e+27) {
		tmp = 2.0 * Math.sqrt(((x * (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 <= -5.6e+36:
		tmp = 2.0 * (1.0 / math.exp((-0.5 * (math.log((-y - z)) - math.log((-1.0 / x))))))
	elif y <= 4.7e+27:
		tmp = 2.0 * math.sqrt(((x * (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 <= -5.6e+36)
		tmp = Float64(2.0 * Float64(1.0 / exp(Float64(-0.5 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x)))))));
	elseif (y <= 4.7e+27)
		tmp = Float64(2.0 * sqrt(Float64(Float64(x * 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 <= -5.6e+36)
		tmp = 2.0 * (1.0 / exp((-0.5 * (log((-y - z)) - log((-1.0 / x))))));
	elseif (y <= 4.7e+27)
		tmp = 2.0 * sqrt(((x * (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, -5.6e+36], N[(2.0 * N[(1.0 / N[Exp[N[(-0.5 * N[(N[Log[N[((-y) - z), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+27], N[(2.0 * N[Sqrt[N[(N[(x * N[(y + z), $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	19.5
Target	11.4
Herbie	3.8

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

Derivation

1. Split input into 3 regimes

2. if y < -5.6000000000000001e36

   1. Initial program 43.5

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

   2. Simplified 43.5

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

      [Start] 43.5	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 43.5	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Applied egg-rr 43.5

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}}} \]

   4. Applied egg-rr 43.5

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

   5. Simplified 43.5

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{{\left(\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)\right)}^{-0.5}}} \]
      Proof

      [Start] 43.5	\[ 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(x, y + z, y \cdot z\right)\right)}^{-0.5}} \]
      fma-def [<=] 43.5	\[ 2 \cdot \frac{1}{{\color{blue}{\left(x \cdot \left(y + z\right) + y \cdot z\right)}}^{-0.5}} \]
      +-commutative [=>] 43.5	\[ 2 \cdot \frac{1}{{\color{blue}{\left(y \cdot z + x \cdot \left(y + z\right)\right)}}^{-0.5}} \]
      fma-def [=>] 43.5	\[ 2 \cdot \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}}^{-0.5}} \]
      *-commutative [=>] 43.5	\[ 2 \cdot \frac{1}{{\left(\mathsf{fma}\left(y, z, \color{blue}{\left(y + z\right) \cdot x}\right)\right)}^{-0.5}} \]

   6. Taylor expanded in x around -inf 6.6

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

   if -5.6000000000000001e36 < y < 4.69999999999999976e27

   1. Initial program 3.8

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

   2. Simplified 3.8

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

      [Start] 3.8	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 3.8	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   if 4.69999999999999976e27 < y

   1. Initial program 41.3

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

   2. Simplified 41.3

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

      [Start] 41.3	\[ 2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
      distribute-lft-out [=>] 41.3	\[ 2 \cdot \sqrt{\color{blue}{x \cdot \left(y + z\right)} + y \cdot z} \]

   3. Taylor expanded in x around 0 41.6

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

   4. Applied egg-rr 1.3

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \frac{1}{e^{-0.5 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.8
Cost	20164

\[\begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+38}:\\ \;\;\;\;2 \cdot e^{-0.5 \cdot \left(\log \left(\frac{-1}{x}\right) + \log \left(\frac{-1}{y + z}\right)\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	11.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-254}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y + z}}{x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	19.2
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-291}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Error	19.1
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{y + z}}{x}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{z}}{y + x}\right)}^{-0.5}\\ \end{array} \]

Alternative 5
Error	19.5
Cost	7104

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

Alternative 6
Error	20.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.28 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	19.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-295}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Error	21.0
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(y \cdot z\right)}^{0.5}\\ \end{array} \]

Alternative 9
Error	21.0
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 10
Error	41.6
Cost	6720

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

Reproduce

herbie shell --seed 2023187 
(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)))))