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

Average Error: 20.4 → 1.3

Time: 12.7s

Precision: binary64

Cost: 13700

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \frac{1}{\frac{\sqrt{\frac{-1}{y}}}{\sqrt{-\left(z + x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0)
   (* 2.0 (/ 1.0 (/ (sqrt (/ -1.0 y)) (sqrt (- (+ z x))))))
   (* 2.0 (* (sqrt (+ y x)) (sqrt z)))))

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 <= 0.0) {
		tmp = 2.0 * (1.0 / (sqrt((-1.0 / y)) / sqrt(-(z + x))));
	} else {
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	}
	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 <= 0.0d0) then
        tmp = 2.0d0 * (1.0d0 / (sqrt(((-1.0d0) / y)) / sqrt(-(z + x))))
    else
        tmp = 2.0d0 * (sqrt((y + x)) * sqrt(z))
    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 <= 0.0) {
		tmp = 2.0 * (1.0 / (Math.sqrt((-1.0 / y)) / Math.sqrt(-(z + x))));
	} else {
		tmp = 2.0 * (Math.sqrt((y + x)) * Math.sqrt(z));
	}
	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 <= 0.0:
		tmp = 2.0 * (1.0 / (math.sqrt((-1.0 / y)) / math.sqrt(-(z + x))))
	else:
		tmp = 2.0 * (math.sqrt((y + x)) * math.sqrt(z))
	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 <= 0.0)
		tmp = Float64(2.0 * Float64(1.0 / Float64(sqrt(Float64(-1.0 / y)) / sqrt(Float64(-Float64(z + x))))));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(y + x)) * sqrt(z)));
	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 <= 0.0)
		tmp = 2.0 * (1.0 / (sqrt((-1.0 / y)) / sqrt(-(z + x))));
	else
		tmp = 2.0 * (sqrt((y + x)) * sqrt(z));
	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, 0.0], N[(2.0 * N[(1.0 / N[(N[Sqrt[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-N[(z + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(y + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	20.4
Target	11.8
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if y < 0.0

   1. Initial program 20.7

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

   2. Simplified 20.7

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]
      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)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))))): 2 points increase in error, 2 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))))): 1 points increase in error, 0 points decrease in error

   3. Applied egg-rr 40.8

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

   4. Applied egg-rr 21.0

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

   5. Taylor expanded in y around inf 22.4

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

   6. Simplified 21.5

      \[\leadsto 2 \cdot \frac{1}{\sqrt{\color{blue}{\frac{\frac{1}{y}}{z + x}}}} \]
      Proof
      (/.f64 (/.f64 1 y) (+.f64 z x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 y (+.f64 z x)))): 27 points increase in error, 36 points decrease in error

   7. Applied egg-rr 2.3

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

   if 0.0 < y

   1. Initial program 20.0

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

   2. Simplified 20.0

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]
      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)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))))): 2 points increase in error, 2 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x y) (*.f64 x z)) (*.f64 y z))))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in z around inf 20.1

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

   4. Applied egg-rr 0.4

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \frac{1}{\frac{\sqrt{\frac{-1}{y}}}{\sqrt{-\left(z + x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	14664

\[\begin{array}{l} t_0 := \left(y \cdot x + z \cdot x\right) + y \cdot z\\ t_1 := 2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x + z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.1
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{x}}{y + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{y + x} \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 3
Error	20.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \frac{1}{\sqrt{\frac{\frac{1}{x}}{y + z}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + z \cdot x}\\ \end{array} \]

Alternative 4
Error	20.4
Cost	7108

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + z \cdot x}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	7104

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

Alternative 6
Error	21.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-271}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 7
Error	20.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 8
Error	21.9
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 9
Error	61.9
Cost	6720

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

Alternative 10
Error	41.7
Cost	6720

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

Reproduce

herbie shell --seed 2022297 
(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)))))