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

Average Error: 20.5 → 1.5

Time: 15.0s

Precision: binary64

Cost: 20164

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{x}\right)}^{-0.25} \cdot {\left(\left(-z\right) - y\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0)
   (* 2.0 (pow (* (pow (/ -1.0 x) -0.25) (pow (- (- z) y) 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 <= 0.0) {
		tmp = 2.0 * pow((pow((-1.0 / x), -0.25) * pow((-z - y), 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 <= 0.0d0) then
        tmp = 2.0d0 * (((((-1.0d0) / x) ** (-0.25d0)) * ((-z - y) ** 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 <= 0.0) {
		tmp = 2.0 * Math.pow((Math.pow((-1.0 / x), -0.25) * Math.pow((-z - y), 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 <= 0.0:
		tmp = 2.0 * math.pow((math.pow((-1.0 / x), -0.25) * math.pow((-z - y), 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 <= 0.0)
		tmp = Float64(2.0 * (Float64((Float64(-1.0 / x) ^ -0.25) * (Float64(Float64(-z) - y) ^ 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 <= 0.0)
		tmp = 2.0 * ((((-1.0 / x) ^ -0.25) * ((-z - y) ^ 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, 0.0], N[(2.0 * N[Power[N[(N[Power[N[(-1.0 / x), $MachinePrecision], -0.25], $MachinePrecision] * N[Power[N[((-z) - y), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	20.5
Target	11.5
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if y < 0.0

   1. Initial program 20.1

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

   2. Applied egg-rr 20.3

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

   3. Taylor expanded in x around -inf 5.9

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

   4. Taylor expanded in x around 0 64.0

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

   5. Simplified 0.8

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\frac{-1}{x}\right)}^{-0.25} \cdot {\left(\left(-z\right) - y\right)}^{0.25}\right)}^{2}} \]
      Proof
      (pow.f64 (*.f64 (pow.f64 (/.f64 -1 x) -1/4) (pow.f64 (-.f64 (neg.f64 z) y) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (pow.f64 (/.f64 -1 x) (Rewrite<= metadata-eval (*.f64 -1 1/4))) (pow.f64 (-.f64 (neg.f64 z) y) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 x)) (*.f64 -1 1/4)))) (pow.f64 (-.f64 (neg.f64 z) y) 1/4)) 2): 63 points increase in error, 75 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (log.f64 (/.f64 -1 x)) -1) 1/4))) (pow.f64 (-.f64 (neg.f64 z) y) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x)))) 1/4)) (pow.f64 (-.f64 (neg.f64 z) y) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (pow.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) y) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (pow.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 y))) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (pow.f64 (+.f64 (*.f64 -1 z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (pow.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 z y))) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (pow.f64 (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 y z))) 1/4)) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 -1 (+.f64 y z))) 1/4)))) 2): 60 points increase in error, 74 points decrease in error
      (pow.f64 (*.f64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4)) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z))))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (Rewrite=> prod-exp_binary64 (exp.f64 (+.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4) (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z))))))) 2): 40 points increase in error, 60 points decrease in error
      (pow.f64 (exp.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) 1/4 (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z))))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (fma.f64 (*.f64 -1 (Rewrite=> log-div_binary64 (-.f64 (log.f64 -1) (log.f64 x)))) 1/4 (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z)))))) 2): 143 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (fma.f64 (*.f64 -1 (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 x))))) 1/4 (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z)))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (fma.f64 (*.f64 -1 (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 x))))) 1/4 (*.f64 1/4 (log.f64 (*.f64 -1 (+.f64 y z)))))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (fma.f64 (*.f64 -1 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 x)))) 1/4 (Rewrite=> *-commutative_binary64 (*.f64 (log.f64 (*.f64 -1 (+.f64 y z))) 1/4)))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 -1 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 x)))) 1/4) (*.f64 (log.f64 (*.f64 -1 (+.f64 y z))) 1/4)))) 2): 0 points increase in error, 0 points decrease in error
      (pow.f64 (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/4 (+.f64 (*.f64 -1 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 x)))) (log.f64 (*.f64 -1 (+.f64 y z))))))) 2): 0 points increase in error, 0 points decrease in error

   if 0.0 < y

   1. Initial program 20.9

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

   2. Applied egg-rr 21.4

      \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}\right)}^{1.5}} \]

   3. Taylor expanded in x around 0 22.2

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

   4. Simplified 22.2

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

   5. 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 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{x}\right)}^{-0.25} \cdot {\left(\left(-z\right) - y\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	20036

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\frac{-1}{x}\right)}^{-0.25} \cdot {\left(-y\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 2
Error	11.2
Cost	14148

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot z + y \cdot x\right) + y \cdot z \leq 10^{+308}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	11.2
Cost	13892

\[\begin{array}{l} t_0 := \left(x \cdot z + y \cdot x\right) + y \cdot z\\ \mathbf{if}\;t_0 \leq 10^{+308}:\\ \;\;\;\;2 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{z} \cdot \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Error	20.5
Cost	7232

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

Alternative 5
Error	21.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{y \cdot z}\\ \end{array} \]

Alternative 6
Error	20.5
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 7
Error	21.8
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 8
Error	42.2
Cost	6720

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

Reproduce

herbie shell --seed 2022300 
(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)))))