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

Average Error: 20.0 → 1.4

Time: 11.7s

Precision: binary64

Cost: 13572

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(-y\right) - z}}{\sqrt{\frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.45e-281)
   (* 2.0 (/ (sqrt (- (- y) z)) (sqrt (/ -1.0 x))))
   (* 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 <= 1.45e-281) {
		tmp = 2.0 * (sqrt((-y - z)) / sqrt((-1.0 / x)));
	} 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 <= 1.45d-281) then
        tmp = 2.0d0 * (sqrt((-y - z)) / sqrt(((-1.0d0) / x)))
    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 <= 1.45e-281) {
		tmp = 2.0 * (Math.sqrt((-y - z)) / Math.sqrt((-1.0 / x)));
	} 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 <= 1.45e-281:
		tmp = 2.0 * (math.sqrt((-y - z)) / math.sqrt((-1.0 / x)))
	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 <= 1.45e-281)
		tmp = Float64(2.0 * Float64(sqrt(Float64(Float64(-y) - z)) / sqrt(Float64(-1.0 / x))));
	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 <= 1.45e-281)
		tmp = 2.0 * (sqrt((-y - z)) / sqrt((-1.0 / x)));
	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, 1.45e-281], N[(2.0 * N[(N[Sqrt[N[((-y) - z), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	20.0
Target	11.6
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if y < 1.44999999999999995e-281

   1. Initial program 20.1

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

   2. Simplified 20.1

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}} \]
      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)))): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 20.3

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

   4. Applied egg-rr 33.7

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

   5. Simplified 27.7

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\frac{z + y}{\frac{y - z}{x}} \cdot \left(y - z\right)}} \]
      Proof
      (*.f64 2 (sqrt.f64 (*.f64 (/.f64 (+.f64 z y) (/.f64 (-.f64 y z) x)) (-.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 y z)) (/.f64 (-.f64 y z) x)) (-.f64 y z)))): 2 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 y z) (-.f64 y z)) (/.f64 (-.f64 y z) x))))): 0 points increase in error, 2 points decrease in error
      (*.f64 2 (sqrt.f64 (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 y y) (*.f64 z z))) (/.f64 (-.f64 y z) x)))): 0 points increase in error, 6 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (*.f64 y y) (*.f64 z z)) x) (-.f64 y z))))): 6 points increase in error, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (-.f64 (*.f64 y y) (*.f64 z z)))) (-.f64 y z)))): 0 points increase in error, 2 points decrease in error

   6. Applied egg-rr 20.5

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

   7. Applied egg-rr 0.8

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

   8. Simplified 0.7

      \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{\left(-y\right) - z}}{\sqrt{\frac{-1}{x}}}} \]
      Proof
      (*.f64 2 (/.f64 (sqrt.f64 (-.f64 (neg.f64 y) z)) (sqrt.f64 (/.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (sqrt.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 y) (neg.f64 z)))) (sqrt.f64 (/.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (sqrt.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 y z)))) (sqrt.f64 (/.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (sqrt.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 z y)))) (sqrt.f64 (/.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (sqrt.f64 (neg.f64 (+.f64 z y))))) (sqrt.f64 (/.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (/.f64 -1 x))) (sqrt.f64 (neg.f64 (+.f64 z y)))))): 0 points increase in error, 0 points decrease in error

   if 1.44999999999999995e-281 < y

   1. Initial program 19.8

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

   2. Simplified 19.8

      \[\leadsto \color{blue}{2 \cdot \sqrt{x \cdot \left(y + z\right) + y \cdot z}} \]
      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)))): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 21.2

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

   4. Applied egg-rr 2.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left(-y\right) - z}}{\sqrt{\frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.6
Cost	14921

\[\begin{array}{l} t_0 := \left(y \cdot x + z \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \end{array} \]

Alternative 2
Error	10.6
Cost	14665

\[\begin{array}{l} t_0 := \left(y \cdot x + z \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \end{array} \]

Alternative 3
Error	19.7
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-281}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{1}{x}}{y + z}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	7104

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

Alternative 5
Error	20.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	20.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-288}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	21.5
Cost	6852

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

Alternative 8
Error	42.0
Cost	6720

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

Reproduce

herbie shell --seed 2022343 
(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)))))