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

Average Error: 19.7 → 3.7

Time: 12.3s

Precision: binary64

Cost: 26564

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -750000000000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 0.00022:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -750000000000.0)
   (* 2.0 (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))
   (if (<= y 0.00022)
     (* 2.0 (sqrt (+ (* y z) (* x (+ 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 <= -750000000000.0) {
		tmp = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	} else if (y <= 0.00022) {
		tmp = 2.0 * sqrt(((y * z) + (x * (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 <= (-750000000000.0d0)) then
        tmp = 2.0d0 * (exp((0.25d0 * (log((-y - z)) - log(((-1.0d0) / x))))) ** 2.0d0)
    else if (y <= 0.00022d0) then
        tmp = 2.0d0 * sqrt(((y * z) + (x * (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 <= -750000000000.0) {
		tmp = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-y - z)) - Math.log((-1.0 / x))))), 2.0);
	} else if (y <= 0.00022) {
		tmp = 2.0 * Math.sqrt(((y * z) + (x * (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 <= -750000000000.0:
		tmp = 2.0 * math.pow(math.exp((0.25 * (math.log((-y - z)) - math.log((-1.0 / x))))), 2.0)
	elif y <= 0.00022:
		tmp = 2.0 * math.sqrt(((y * z) + (x * (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 <= -750000000000.0)
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0));
	elseif (y <= 0.00022)
		tmp = Float64(2.0 * sqrt(Float64(Float64(y * z) + Float64(x * 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 <= -750000000000.0)
		tmp = 2.0 * (exp((0.25 * (log((-y - z)) - log((-1.0 / x))))) ^ 2.0);
	elseif (y <= 0.00022)
		tmp = 2.0 * sqrt(((y * z) + (x * (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, -750000000000.0], 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, 0.00022], N[(2.0 * N[Sqrt[N[(N[(y * z), $MachinePrecision] + N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.7
Target	11.4
Herbie	3.7

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

Derivation

1. Split input into 3 regimes

2. if y < -7.5e11

   1. Initial program 38.4

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

   2. Simplified 38.4

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

   3. Applied egg-rr 38.6

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

   4. Taylor expanded in x around -inf 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 -7.5e11 < y < 2.20000000000000008e-4

   1. Initial program 3.5

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

   2. Simplified 3.5

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

   if 2.20000000000000008e-4 < y

   1. Initial program 34.5

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

   2. Simplified 34.5

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]
      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)))))): 2 points increase in error, 3 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))))): 0 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. Applied egg-rr 56.4

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

   4. Simplified 56.4

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{y \cdot \left(y \cdot \left(x \cdot x\right)\right) - {\left(z \cdot \left(y + x\right)\right)}^{2}}{y \cdot \left(x - z\right) - z \cdot x}}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y (*.f64 y (*.f64 x x))) (pow.f64 (*.f64 z (+.f64 y x)) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y x) x))) (pow.f64 (*.f64 z (+.f64 y x)) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 9 points increase in error, 13 points decrease in error
      (/.f64 (-.f64 (*.f64 y (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) x)) (pow.f64 (*.f64 z (+.f64 y x)) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 x y) x) y)) (pow.f64 (*.f64 z (+.f64 y x)) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 x y) (*.f64 x y))) (pow.f64 (*.f64 z (+.f64 y x)) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 6 points increase in error, 14 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (Rewrite<= +-commutative_binary64 (+.f64 x y))) 2)) (-.f64 (*.f64 y (-.f64 x z)) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (-.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x y) (*.f64 z y))) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (-.f64 (-.f64 (*.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 y z))) (*.f64 z x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (-.f64 (-.f64 (*.f64 x y) (*.f64 y z)) (Rewrite=> *-commutative_binary64 (*.f64 x z)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x y) (+.f64 (*.f64 y z) (*.f64 x z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (-.f64 (*.f64 x y) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x z) (*.f64 y z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 (*.f64 x y) (*.f64 x y)) (pow.f64 (*.f64 z (+.f64 x y)) 2)) (-.f64 (*.f64 x y) (Rewrite=> distribute-rgt-out_binary64 (*.f64 z (+.f64 x y))))): 2 points increase in error, 2 points decrease in error

   5. Taylor expanded in x around 0 35.1

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

   6. Simplified 35.1

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

   7. Applied egg-rr 1.4

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750000000000:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 0.00022:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	11.2
Cost	13892

\[\begin{array}{l} \mathbf{if}\;y \cdot z + \left(y \cdot x + x \cdot z\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	19.7
Cost	7104

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

Alternative 3
Error	21.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-243}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Error	19.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 10^{-273}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5
Error	21.2
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 6
Error	42.3
Cost	6720

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

Reproduce

herbie shell --seed 2022325 
(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)))))