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

Average Accuracy: 90.6% → 98.0%

Time: 14.6s

Precision: binary64

Cost: 7624

Math

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

↓

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

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 (/ (* y (- t z)) a)) (fma y (/ (- t z) 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 + ((y * (t - z)) / a);
	} else {
		tmp = fma(y, ((t - z) / 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(Float64(y * Float64(t - z)) / a));
	else
		tmp = fma(y, Float64(Float64(t - z) / 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[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(t - z), $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%

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

      [Start] 74.4	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 74.4	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 74.4	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 74.4	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 97.4	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 97.4	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 94.2	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 94.2	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 94.2	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 94.2	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 94.2	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 94.2	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 94.2	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 94.2	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	1353

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

Alternative 2
Accuracy	81.0%
Cost	1109

\[\begin{array}{l} t_1 := x - y \cdot \frac{z}{a}\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+28}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+139} \lor \neg \left(z \leq 1.35 \cdot 10^{+192}\right):\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 3
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 4
Accuracy	55.7%
Cost	980

\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	55.7%
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-220}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-286}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-121} \lor \neg \left(x \leq 2.1 \cdot 10^{-243} \lor \neg \left(x \leq 2.8 \cdot 10^{-193}\right) \land x \leq 1.22 \cdot 10^{-35}\right):\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 7
Accuracy	81.4%
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+17} \lor \neg \left(z \leq 2.35 \cdot 10^{+30} \lor \neg \left(z \leq 1.9 \cdot 10^{+99}\right) \land z \leq 7.8 \cdot 10^{+145}\right):\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 8
Accuracy	81.7%
Cost	977

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+21}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{+137} \lor \neg \left(z \leq 1.2 \cdot 10^{+169}\right):\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}}\\ \end{array} \]

Alternative 9
Accuracy	51.4%
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+121} \lor \neg \left(y \leq -1.35 \cdot 10^{+48}\right) \land \left(y \leq -25000000000 \lor \neg \left(y \leq 5.2 \cdot 10^{+41}\right)\right):\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	51.1%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -11400000000 \lor \neg \left(y \leq 2.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	75.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-154} \lor \neg \left(x \leq 1.75 \cdot 10^{-143}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 12
Accuracy	68.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	96.3%
Cost	576

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

Alternative 14
Accuracy	52.3%
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, F"
  :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)))