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

Average Error: 5.9 → 1.4

Time: 13.5s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-59}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.05e-115)
   (fma y (/ (- z t) a) x)
   (if (<= a 9e-59)
     (+ x (/ 1.0 (/ a (* y (- z t)))))
     (+ 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 tmp;
	if (a <= -1.05e-115) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (a <= 9e-59) {
		tmp = x + (1.0 / (a / (y * (z - t))));
	} 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)
	tmp = 0.0
	if (a <= -1.05e-115)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (a <= 9e-59)
		tmp = Float64(x + Float64(1.0 / Float64(a / Float64(y * Float64(z - t)))));
	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_] := If[LessEqual[a, -1.05e-115], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 9e-59], N[(x + N[(1.0 / N[(a / N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	5.9
Target	0.8
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if a < -1.05000000000000001e-115

   1. Initial program 7.1

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

   2. Simplified 2.0

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

      [Start] 7.1	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 7.1	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 2.0	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 2.0	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -1.05000000000000001e-115 < a < 9.00000000000000023e-59

   1. Initial program 1.0

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

   2. Simplified 4.9

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

      [Start] 1.0	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 4.9	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   3. Applied egg-rr 1.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{a}{y \cdot \left(z - t\right)}}} \]

   if 9.00000000000000023e-59 < a

   1. Initial program 8.1

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

   2. Simplified 1.1

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

      [Start] 8.1	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 1.1	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-59}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	1373

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{a}\\ \mathbf{if}\;z \leq -10000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-191}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-16} \lor \neg \left(z \leq 5.6 \cdot 10^{+74}\right) \land z \leq 3 \cdot 10^{+117}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+279}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+197}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Error	30.3
Cost	1244

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-253}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-133}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	30.3
Cost	1244

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-243}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-135}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-77}:\\ \;\;\;\;-\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	12.8
Cost	1241

\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{a}\\ t_2 := x + \frac{y \cdot z}{a}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+167} \lor \neg \left(z \leq 1.9 \cdot 10^{+229}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 6
Error	1.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -1 \cdot 10^{+81} \lor \neg \left(z - t \leq 10^{+34}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 7
Error	30.5
Cost	980

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-243}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.18 \cdot 10^{-100}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	30.5
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-244}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	21.4
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-53} \lor \neg \left(x \leq -2.3 \cdot 10^{-92}\right) \land x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	1.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-134} \lor \neg \left(a \leq 10^{-62}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \end{array} \]

Alternative 11
Error	17.9
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+202}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+108}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+180}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \end{array} \]

Alternative 12
Error	10.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+55} \lor \neg \left(t \leq 2.15 \cdot 10^{+21}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \]

Alternative 13
Error	29.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	30.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	30.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-126}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	2.6
Cost	576

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

Alternative 17
Error	31.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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)))