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

Average Error: 6.2 → 1.5

Time: 13.5s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -2.2e+80)
     (fma y (/ (- z t) a) x)
     (if (<= t_1 2e+111) (+ x (/ t_1 a)) (+ x (/ y (/ a (- z t))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -2.2e+80) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+111) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -2.2e+80)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+111)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2.2e+80], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+111], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.2
Target	0.7
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -2.20000000000000003e80

   1. Initial program 14.9

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

   2. Simplified 3.9

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

      [Start] 14.9	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 14.9	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 3.9	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 3.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -2.20000000000000003e80 < (*.f64 y (-.f64 z t)) < 1.99999999999999991e111

   1. Initial program 0.5

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

   if 1.99999999999999991e111 < (*.f64 y (-.f64 z t))

   1. Initial program 18.4

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

   2. Simplified 2.5

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]
      Proof

      [Start] 18.4	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 2.5	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	33.7
Cost	1440

\[\begin{array}{l} t_1 := y \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -5 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-281}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 580000000:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	33.4
Cost	1440

\[\begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+110}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-282}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 195000000:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+137}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+185}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	33.4
Cost	1440

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{+57}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-281}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 126000000:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+136}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 10^{+187}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	22.1
Cost	1373

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a}\\ \mathbf{if}\;x \leq -30500000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-163}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-89} \lor \neg \left(x \leq 360000\right) \land x \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.8
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+283}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a}\\ t_2 := x + \frac{z}{\frac{a}{y}}\\ t_3 := x + \frac{y \cdot z}{a}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-272}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 200000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	16.5
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-163} \lor \neg \left(x \leq 3.6 \cdot 10^{-131}\right) \land \left(x \leq 175000000000 \lor \neg \left(x \leq 5.4 \cdot 10^{+53}\right)\right):\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-165} \lor \neg \left(x \leq 4.6 \cdot 10^{-133}\right) \land \left(x \leq 200000000000 \lor \neg \left(x \leq 6.2 \cdot 10^{+53}\right)\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 9
Error	10.2
Cost	976

\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{a}\\ t_2 := x + \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-94}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	2.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+23} \lor \neg \left(z \leq 4 \cdot 10^{+18}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 11
Error	27.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-90}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.4
Cost	576

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

Alternative 13
Error	30.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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