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

Average Error: 19.5 → 4.4

Time: 6.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(1, z \cdot \left(y + x\right), y \cdot x\right)}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot \left({1}^{0.5} \cdot \sqrt{\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (* 2.0 (pow (exp (* 0.25 (- (log (+ y x)) (log (/ 1.0 z))))) 2.0)))
        (t_1
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -8e+17)
     t_1
     (if (<= y -9e-266)
       (* 2.0 (sqrt (fma 1.0 (* z (+ y x)) (* y x))))
       (if (<= y 6.2e-287)
         t_1
         (if (<= y 2e-186)
           t_0
           (if (<= y 7e+35)
             (* 2.0 (* (pow 1.0 0.5) (sqrt (fma y z (* (+ y z) x)))))
             t_0)))))))

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 t_0 = 2.0 * pow(exp((0.25 * (log((y + x)) - log((1.0 / z))))), 2.0);
	double t_1 = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -8e+17) {
		tmp = t_1;
	} else if (y <= -9e-266) {
		tmp = 2.0 * sqrt(fma(1.0, (z * (y + x)), (y * x)));
	} else if (y <= 6.2e-287) {
		tmp = t_1;
	} else if (y <= 2e-186) {
		tmp = t_0;
	} else if (y <= 7e+35) {
		tmp = 2.0 * (pow(1.0, 0.5) * sqrt(fma(y, z, ((y + z) * x))));
	} else {
		tmp = t_0;
	}
	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)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(y + x)) - log(Float64(1.0 / z))))) ^ 2.0))
	t_1 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -8e+17)
		tmp = t_1;
	elseif (y <= -9e-266)
		tmp = Float64(2.0 * sqrt(fma(1.0, Float64(z * Float64(y + x)), Float64(y * x))));
	elseif (y <= 6.2e-287)
		tmp = t_1;
	elseif (y <= 2e-186)
		tmp = t_0;
	elseif (y <= 7e+35)
		tmp = Float64(2.0 * Float64((1.0 ^ 0.5) * sqrt(fma(y, z, Float64(Float64(y + z) * x)))));
	else
		tmp = t_0;
	end
	return 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_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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, -8e+17], t$95$1, If[LessEqual[y, -9e-266], N[(2.0 * N[Sqrt[N[(1.0 * N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-287], t$95$1, If[LessEqual[y, 2e-186], t$95$0, If[LessEqual[y, 7e+35], N[(2.0 * N[(N[Power[1.0, 0.5], $MachinePrecision] * N[Sqrt[N[(y * z + N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	19.5
Target	11.2
Herbie	4.4

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

Derivation

1. Split input into 4 regimes

2. if y < -8e17 or -9.0000000000000006e-266 < y < 6.2000000000000001e-287

   1. Initial program 39.0

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

   2. Applied egg-rr 39.2

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

   3. Taylor expanded in x around -inf 7.0

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

   if -8e17 < y < -9.0000000000000006e-266

   1. Initial program 2.9

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

   2. Applied egg-rr 2.9

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

   if 6.2000000000000001e-287 < y < 1.9999999999999998e-186 or 7.0000000000000001e35 < y

   1. Initial program 37.2

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

   2. Applied egg-rr 37.4

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

   3. Taylor expanded in z around inf 6.4

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

   if 1.9999999999999998e-186 < y < 7.0000000000000001e35

   1. Initial program 1.3

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

   2. Applied egg-rr 1.7

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

   3. Applied egg-rr 1.3

      \[\leadsto 2 \cdot \color{blue}{\left({1}^{0.5} \cdot \sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-266}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(1, z \cdot \left(y + x\right), y \cdot x\right)}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-186}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot \left({1}^{0.5} \cdot \sqrt{\mathsf{fma}\left(y, z, \left(y + z\right) \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))))