Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Average Error: 6.5 → 1.8

Time: 23.5s

Precision: binary64

Cost: 9616

Math

\[x + \frac{y \cdot \left(z - x\right)}{t} \]

↓

\[\begin{array}{l} t_1 := x + \frac{\mathsf{fma}\left(z, y, -x \cdot y\right)}{t}\\ t_2 := x + \frac{y \cdot \left(z - x\right)}{t}\\ t_3 := x + \frac{z - x}{t} \cdot y\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (* y (- z x)) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (fma z y (- (* x y))) t)))
        (t_2 (+ x (/ (* y (- z x)) t)))
        (t_3 (+ x (* (/ (- z x) t) y))))
   (if (<= t_2 (- INFINITY))
     (* (/ y t) (- z x))
     (if (<= t_2 -4e+15)
       t_1
       (if (<= t_2 2e-134) t_3 (if (<= t_2 2e+302) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = x + (fma(z, y, -(x * y)) / t);
	double t_2 = x + ((y * (z - x)) / t);
	double t_3 = x + (((z - x) / t) * y);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (y / t) * (z - x);
	} else if (t_2 <= -4e+15) {
		tmp = t_1;
	} else if (t_2 <= 2e-134) {
		tmp = t_3;
	} else if (t_2 <= 2e+302) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * Float64(z - x)) / t))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x + Float64(fma(z, y, Float64(-Float64(x * y))) / t))
	t_2 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	t_3 = Float64(x + Float64(Float64(Float64(z - x) / t) * y))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z - x));
	elseif (t_2 <= -4e+15)
		tmp = t_1;
	elseif (t_2 <= 2e-134)
		tmp = t_3;
	elseif (t_2 <= 2e+302)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(z * y + (-N[(x * y), $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e+15], t$95$1, If[LessEqual[t$95$2, 2e-134], t$95$3, If[LessEqual[t$95$2, 2e+302], t$95$1, t$95$3]]]]]]]

Target

Original	6.5
Target	2.0
Herbie	1.8

\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right) \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0

   1. Initial program 64.0

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]

   2. Taylor expanded in y around inf 20.5

      \[\leadsto \color{blue}{y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)} \]

   3. Applied egg-rr 20.5

      \[\leadsto \color{blue}{\frac{y}{t} \cdot \left(z - x\right)} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -4e15 or 2.00000000000000008e-134 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 2.0000000000000002e302

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]

   2. Applied egg-rr 0.2

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

   if -4e15 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 2.00000000000000008e-134 or 2.0000000000000002e302 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 11.9

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]

   2. Applied egg-rr 2.5

      \[\leadsto x + \color{blue}{\frac{z - x}{t} \cdot y} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	1.9
Cost	3280

\[\begin{array}{l} t_1 := x + \frac{z - x}{t} \cdot y\\ t_2 := y \cdot \left(z - x\right)\\ t_3 := x + \frac{t_2}{t}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{elif}\;t_3 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;x + \frac{1}{t} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.9
Cost	3152

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ t_2 := x + \frac{z - x}{t} \cdot y\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \left(z - x\right)\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 20000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	30.6
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ \mathbf{if}\;z \leq -8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+180}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-172}:\\ \;\;\;\;-\frac{y \cdot x}{t}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+89}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.6
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{+179}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -12.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \left(-\frac{x}{t}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.2
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ t_2 := \left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{if}\;z \leq -8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -30:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.5
Cost	1240

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \left(z - x\right)\\ t_2 := \left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+226}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -15:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.92 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	10.1
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{t} \cdot z\\ t_2 := \left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+86}:\\ \;\;\;\;\frac{t - y}{t} \cdot x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	29.8
Cost	1112

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ \mathbf{if}\;z \leq -8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+181}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -12.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	12.7
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{t} \cdot z\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	16.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;\frac{t - y}{t} \cdot x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \end{array} \]

Alternative 11
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+246}:\\ \;\;\;\;y \cdot \left(\frac{z}{t} - \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \]

Alternative 12
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{z - x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot \left(z - x\right)\\ \end{array} \]

Alternative 13
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))