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

Average Error: 19.9 → 4.3

Time: 5.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ t_2 := 2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (- (- z) y)) (log (/ -1.0 x))))) 2.0)))
        (t_2 (* 2.0 (sqrt (fma x y (* z (+ y x)))))))
   (if (<= y -7e+52)
     t_1
     (if (<= y -1.25e-221)
       t_2
       (if (<= y 0.0) t_1 (if (<= y 1e-200) t_0 (if (<= y 1e+50) t_2 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((-z - y)) - log((-1.0 / x))))), 2.0);
	double t_2 = 2.0 * sqrt(fma(x, y, (z * (y + x))));
	double tmp;
	if (y <= -7e+52) {
		tmp = t_1;
	} else if (y <= -1.25e-221) {
		tmp = t_2;
	} else if (y <= 0.0) {
		tmp = t_1;
	} else if (y <= 1e-200) {
		tmp = t_0;
	} else if (y <= 1e+50) {
		tmp = t_2;
	} 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(-z) - y)) - log(Float64(-1.0 / x))))) ^ 2.0))
	t_2 = Float64(2.0 * sqrt(fma(x, y, Float64(z * Float64(y + x)))))
	tmp = 0.0
	if (y <= -7e+52)
		tmp = t_1;
	elseif (y <= -1.25e-221)
		tmp = t_2;
	elseif (y <= 0.0)
		tmp = t_1;
	elseif (y <= 1e-200)
		tmp = t_0;
	elseif (y <= 1e+50)
		tmp = t_2;
	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[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[N[(x * y + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+52], t$95$1, If[LessEqual[y, -1.25e-221], t$95$2, If[LessEqual[y, 0.0], t$95$1, If[LessEqual[y, 1e-200], t$95$0, If[LessEqual[y, 1e+50], t$95$2, t$95$0]]]]]]]]

Target

Original	19.9
Target	11.5
Herbie	4.3

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

Derivation

1. Split input into 3 regimes

2. if y < -7e52 or -1.24999999999999999e-221 < y < 0.0

   1. Initial program 42.2

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

   2. Simplified 42.2

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

   3. Applied egg-rr 42.3

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

   4. Taylor expanded in x around -inf 6.5

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

   if -7e52 < y < -1.24999999999999999e-221 or 9.9999999999999998e-201 < y < 1.0000000000000001e50

   1. Initial program 2.5

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

   2. Simplified 2.5

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

   3. Applied egg-rr 2.8

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

   4. Applied egg-rr 2.5

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

   if 0.0 < y < 9.9999999999999998e-201 or 1.0000000000000001e50 < y

   1. Initial program 40.9

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

   2. Simplified 40.9

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

   3. Applied egg-rr 41.0

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

   4. Taylor expanded in z around inf 6.6

      \[\leadsto 2 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log \left(y + x\right)\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-221}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 0:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 10^{-200}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 10^{+50}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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)))))