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

Average Accuracy: 90.6% → 98.0%

Time: 13.5s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+127}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+101}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]

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 -1e+127)
     (+ x (/ y (/ a (- z t))))
     (if (<= t_1 1e+101) (+ x (/ t_1 a)) (fma y (/ (- z t) a) x)))))

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 <= -1e+127) {
		tmp = x + (y / (a / (z - t)));
	} else if (t_1 <= 1e+101) {
		tmp = x + (t_1 / a);
	} else {
		tmp = fma(y, ((z - t) / a), x);
	}
	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 <= -1e+127)
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	elseif (t_1 <= 1e+101)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	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, -1e+127], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+101], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	90.6%
Target	99.1%
Herbie	98.0%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -9.99999999999999955e126

   1. Initial program 70.3%

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

   2. Simplified 96.4%

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

      [Start] 70.3	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 96.4	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   if -9.99999999999999955e126 < (*.f64 y (-.f64 z t)) < 9.9999999999999998e100

   1. Initial program 99.4%

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

   if 9.9999999999999998e100 < (*.f64 y (-.f64 z t))

   1. Initial program 74.4%

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

   2. Simplified 94.2%

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

      [Start] 74.4	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 74.4	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 94.2	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 94.2	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+127}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+101}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+127} \lor \neg \left(t_1 \leq 4 \cdot 10^{+294}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{-48} \lor \neg \left(z - t \leq 2 \cdot 10^{+60}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Accuracy	50.9%
Cost	1044

\[\begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ t_2 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -11500000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+229}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	50.9%
Cost	1044

\[\begin{array}{l} t_1 := y \cdot \frac{z}{a}\\ t_2 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+130}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1150000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	55.9%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-287}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	55.8%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	73.5%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+109} \lor \neg \left(y \leq -3.55 \cdot 10^{+47}\right) \land \left(y \leq -25000000000 \lor \neg \left(y \leq 5.4 \cdot 10^{+41}\right)\right):\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 8
Accuracy	85.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+21} \lor \neg \left(z \leq 5.2 \cdot 10^{+27}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9
Accuracy	69.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0126:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	56.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-144}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	57.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	57.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-144}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	96.3%
Cost	576

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

Alternative 14
Accuracy	52.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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)))