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

Average Error: 20.1 → 2.4

Time: 11.2s

Precision: binary64

Cost: 20036

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{y}\right)}^{-0.25} \cdot {\left(-x\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-310)
   (* 2.0 (pow (* (pow (/ -1.0 y) -0.25) (pow (- x) 0.25)) 2.0))
   (* 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 <= 2e-310) {
		tmp = 2.0 * pow((pow((-1.0 / y), -0.25) * pow(-x, 0.25)), 2.0);
	} 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 <= 2d-310) then
        tmp = 2.0d0 * (((((-1.0d0) / y) ** (-0.25d0)) * (-x ** 0.25d0)) ** 2.0d0)
    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 <= 2e-310) {
		tmp = 2.0 * Math.pow((Math.pow((-1.0 / y), -0.25) * Math.pow(-x, 0.25)), 2.0);
	} 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 <= 2e-310:
		tmp = 2.0 * math.pow((math.pow((-1.0 / y), -0.25) * math.pow(-x, 0.25)), 2.0)
	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 <= 2e-310)
		tmp = Float64(2.0 * (Float64((Float64(-1.0 / y) ^ -0.25) * (Float64(-x) ^ 0.25)) ^ 2.0));
	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 <= 2e-310)
		tmp = 2.0 * ((((-1.0 / y) ^ -0.25) * (-x ^ 0.25)) ^ 2.0);
	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, 2e-310], N[(2.0 * N[Power[N[(N[Power[N[(-1.0 / y), $MachinePrecision], -0.25], $MachinePrecision] * N[Power[(-x), 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	20.1
Target	11.3
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if y < 1.999999999999994e-310

   1. Initial program 19.7

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

   2. Simplified 19.7

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

   3. Taylor expanded in z around 0 21.2

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

   4. Applied egg-rr 21.4

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(y \cdot x\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in y around -inf 7.2

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

   6. Simplified 2.5

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{-1}{y}\right)}^{-0.25} \cdot {\left(-x\right)}^{0.25}\right)}}^{2} \]
      Proof
      (*.f64 (pow.f64 (/.f64 -1 y) -1/4) (pow.f64 (neg.f64 x) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 -1 y) (Rewrite<= metadata-eval (*.f64 -1 1/4))) (pow.f64 (neg.f64 x) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 y)) (*.f64 -1 1/4)))) (pow.f64 (neg.f64 x) 1/4)): 57 points increase in error, 57 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (log.f64 (/.f64 -1 y)) -1) 1/4))) (pow.f64 (neg.f64 x) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (log.f64 (/.f64 -1 y)))) 1/4)) (pow.f64 (neg.f64 x) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 y))) 1/4)) (pow.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 y))) 1/4)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 -1 x)) 1/4)))): 62 points increase in error, 57 points decrease in error
      (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 y))) 1/4) (*.f64 (log.f64 (*.f64 -1 x)) 1/4)))): 35 points increase in error, 52 points decrease in error
      (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/4 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 y))) (log.f64 (*.f64 -1 x)))))): 0 points increase in error, 0 points decrease in error

   if 1.999999999999994e-310 < y

   1. Initial program 20.4

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

   2. Simplified 20.4

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]
      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, 0 points decrease in error
      (*.f64 2 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))))): 3 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 36.0

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   4. Simplified 33.9

      \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(y + z, x, y \cdot z\right)\right)}^{1.5}}} \]
      Proof
      (cbrt.f64 (pow.f64 (fma.f64 (+.f64 y z) x (*.f64 y z)) 3/2)): 0 points increase in error, 0 points decrease in error
      (cbrt.f64 (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 y z) x) (*.f64 y z))) 3/2)): 0 points increase in error, 0 points decrease in error
      (cbrt.f64 (pow.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x (+.f64 y z))) (*.f64 y z)) 3/2)): 0 points increase in error, 0 points decrease in error
      (cbrt.f64 (pow.f64 (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x y) (*.f64 x z))) (*.f64 y z)) 3/2)): 1 points increase in error, 1 points decrease in error
      (cbrt.f64 (pow.f64 (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 x y) (+.f64 (*.f64 x z) (*.f64 y z)))) 3/2)): 0 points increase in error, 0 points decrease in error
      (cbrt.f64 (pow.f64 (+.f64 (*.f64 x y) (Rewrite=> distribute-rgt-out_binary64 (*.f64 z (+.f64 x y)))) 3/2)): 0 points increase in error, 1 points decrease in error
      (cbrt.f64 (pow.f64 (Rewrite<= fma-udef_binary64 (fma.f64 x y (*.f64 z (+.f64 x y)))) 3/2)): 0 points increase in error, 1 points decrease in error
      (Rewrite<= unpow1/3_binary64 (pow.f64 (pow.f64 (fma.f64 x y (*.f64 z (+.f64 x y))) 3/2) 1/3)): 87 points increase in error, 48 points decrease in error

   5. Taylor expanded in x around 0 21.8

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

   6. Simplified 21.8

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

   7. 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 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{y}\right)}^{-0.25} \cdot {\left(-x\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	20036

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{x}\right)}^{-0.25} \cdot {\left(-y\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	3.9
Cost	19972

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+61}:\\ \;\;\;\;2 \cdot e^{\left(\log \left(-x\right) - \log \left(\frac{-1}{y}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-227}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	11.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-227}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z + x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Error	20.1
Cost	7104

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

Alternative 5
Error	21.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6
Error	20.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-270}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 7
Error	21.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-270}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 8
Error	41.8
Cost	6720

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

Reproduce

herbie shell --seed 2022331 
(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)))))