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

Average Error: 20.1 → 5.2

Time: 5.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(y + x\right)\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ t_2 := 2 \cdot \sqrt{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, x, y \cdot z\right)\right)}\\ t_3 := {\left({\left(e^{0.25}\right)}^{\left(t_0 + \log z\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(0.5, \frac{y}{z} \cdot \frac{x}{\frac{y + x}{t_3}}, t_3\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(t_0 - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

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 (log (+ y x)))
        (t_1
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- y) z)) (log (/ -1.0 x))))) 2.0)))
        (t_2 (* 2.0 (sqrt (fma x y (fma z x (* y z))))))
        (t_3 (pow (pow (exp 0.25) (+ t_0 (log z))) 2.0)))
   (if (<= y -1.25e+97)
     t_1
     (if (<= y -6.8e-180)
       t_2
       (if (<= y 1.6e-262)
         t_1
         (if (<= y 4.6e-212)
           (* 2.0 (fma 0.5 (* (/ y z) (/ x (/ (+ y x) t_3))) t_3))
           (if (<= y 5e+47)
             t_2
             (* 2.0 (pow (exp (* 0.25 (- t_0 (log (/ 1.0 z))))) 2.0)))))))))

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 = log((y + x));
	double t_1 = 2.0 * pow(exp((0.25 * (log((-y - z)) - log((-1.0 / x))))), 2.0);
	double t_2 = 2.0 * sqrt(fma(x, y, fma(z, x, (y * z))));
	double t_3 = pow(pow(exp(0.25), (t_0 + log(z))), 2.0);
	double tmp;
	if (y <= -1.25e+97) {
		tmp = t_1;
	} else if (y <= -6.8e-180) {
		tmp = t_2;
	} else if (y <= 1.6e-262) {
		tmp = t_1;
	} else if (y <= 4.6e-212) {
		tmp = 2.0 * fma(0.5, ((y / z) * (x / ((y + x) / t_3))), t_3);
	} else if (y <= 5e+47) {
		tmp = t_2;
	} else {
		tmp = 2.0 * pow(exp((0.25 * (t_0 - log((1.0 / z))))), 2.0);
	}
	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 = log(Float64(y + x))
	t_1 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-y) - z)) - log(Float64(-1.0 / x))))) ^ 2.0))
	t_2 = Float64(2.0 * sqrt(fma(x, y, fma(z, x, Float64(y * z)))))
	t_3 = (exp(0.25) ^ Float64(t_0 + log(z))) ^ 2.0
	tmp = 0.0
	if (y <= -1.25e+97)
		tmp = t_1;
	elseif (y <= -6.8e-180)
		tmp = t_2;
	elseif (y <= 1.6e-262)
		tmp = t_1;
	elseif (y <= 4.6e-212)
		tmp = Float64(2.0 * fma(0.5, Float64(Float64(y / z) * Float64(x / Float64(Float64(y + x) / t_3))), t_3));
	elseif (y <= 5e+47)
		tmp = t_2;
	else
		tmp = Float64(2.0 * (exp(Float64(0.25 * Float64(t_0 - log(Float64(1.0 / z))))) ^ 2.0));
	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[Log[N[(y + x), $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]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[N[(x * y + N[(z * x + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(t$95$0 + N[Log[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -1.25e+97], t$95$1, If[LessEqual[y, -6.8e-180], t$95$2, If[LessEqual[y, 1.6e-262], t$95$1, If[LessEqual[y, 4.6e-212], N[(2.0 * N[(0.5 * N[(N[(y / z), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+47], t$95$2, N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(t$95$0 - N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.1
Target	11.7
Herbie	5.2

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

Derivation

1. Split input into 4 regimes

2. if y < -1.25e97 or -6.79999999999999963e-180 < y < 1.6e-262

   1. Initial program 43.7

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

   2. Applied egg-rr 43.8

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

   3. Taylor expanded in x around -inf 8.4

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

   if -1.25e97 < y < -6.79999999999999963e-180 or 4.6000000000000002e-212 < y < 5.00000000000000022e47

   1. Initial program 3.4

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

   2. Simplified 3.4

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

   3. Applied egg-rr 3.4

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

   if 1.6e-262 < y < 4.6000000000000002e-212

   1. Initial program 18.0

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

   2. Applied egg-rr 18.3

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

   3. Taylor expanded in z around inf 22.6

      \[\leadsto 2 \cdot \color{blue}{\left(0.5 \cdot \frac{y \cdot \left({\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2} \cdot x\right)}{z \cdot \left(y + x\right)} + {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\right)} \]

   4. Simplified 8.4

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{y}{z} \cdot \frac{x}{\frac{x + y}{{\left({\left(e^{0.25}\right)}^{\left(\log \left(x + y\right) + \log z\right)}\right)}^{2}}}, {\left({\left(e^{0.25}\right)}^{\left(\log \left(x + y\right) + \log z\right)}\right)}^{2}\right)} \]

   if 5.00000000000000022e47 < y

   1. Initial program 45.1

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

   2. Applied egg-rr 45.2

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

   3. Taylor expanded in z around inf 6.6

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, x, y \cdot z\right)\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-262}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-y\right) - z\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(0.5, \frac{y}{z} \cdot \frac{x}{\frac{y + x}{{\left({\left(e^{0.25}\right)}^{\left(\log \left(y + x\right) + \log z\right)}\right)}^{2}}}, {\left({\left(e^{0.25}\right)}^{\left(\log \left(y + x\right) + \log z\right)}\right)}^{2}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+47}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(x, y, \mathsf{fma}\left(z, x, y \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))))