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

Average Error: 6.0 → 0.8

Time: 19.9s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))) (t_2 (fma y (/ (- t z) a) x)))
   (if (<= t_1 -2e+130)
     t_2
     (if (<= t_1 2e+158) (+ x (/ (* y (- t z)) a)) t_2))))

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 t_2 = fma(y, ((t - z) / a), x);
	double tmp;
	if (t_1 <= -2e+130) {
		tmp = t_2;
	} else if (t_1 <= 2e+158) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(y, Float64(Float64(t - z) / a), x)
	tmp = 0.0
	if (t_1 <= -2e+130)
		tmp = t_2;
	elseif (t_1 <= 2e+158)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+130], t$95$2, If[LessEqual[t$95$1, 2e+158], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Target

Original	6.0
Target	0.7
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -2.0000000000000001e130 or 1.99999999999999991e158 < (*.f64 y (-.f64 z t))

   1. Initial program 20.1

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

   2. Simplified 1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)} \]
      Proof
      (fma.f64 y (/.f64 (-.f64 t z) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (/.f64 (-.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 t))) z) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (neg.f64 t)) (neg.f64 z))) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 t) z))) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 z (neg.f64 t)))) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 z t))) a) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 y (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (-.f64 z t) a))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 (/.f64 (-.f64 z t) a))) x)): 2 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 y (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (-.f64 z t)) a))) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (neg.f64 (-.f64 z t))) a)) x): 40 points increase in error, 41 points decrease in error
      (+.f64 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y (-.f64 z t)))) a) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 y (-.f64 z t)) a))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (/.f64 (*.f64 y (-.f64 z t)) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))): 0 points increase in error, 0 points decrease in error

   if -2.0000000000000001e130 < (*.f64 y (-.f64 z t)) < 1.99999999999999991e158

   1. Initial program 0.4

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+158}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1608

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+63}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 2
Error	13.8
Cost	1372

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x - z \cdot \frac{y}{a}\\ t_3 := x - \frac{z}{\frac{a}{y}}\\ t_4 := x + t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.48045622450914 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7118062017964185 \cdot 10^{-37}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 1.3888530738330703 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.1264466371616284 \cdot 10^{+105}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.0
Cost	1372

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x - z \cdot \frac{y}{a}\\ t_3 := x - \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.48045622450914 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8477496664605297 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.3888530738330703 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.1264466371616284 \cdot 10^{+105}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	14.0
Cost	1372

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x - \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.48045622450914 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8477496664605297 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.3888530738330703 \cdot 10^{-13}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 3.1264466371616284 \cdot 10^{+105}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+300}:\\ \;\;\;\;x - \frac{\frac{y}{a}}{\frac{1}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+127}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Error	14.1
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 1.3888530738330703 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1264466371616284 \cdot 10^{+105}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7
Error	14.3
Cost	1240

\[\begin{array}{l} t_1 := x - \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.764474580516205 \cdot 10^{-243}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 1.3888530738330703 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1264466371616284 \cdot 10^{+105}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Error	27.9
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.48045622450914 \cdot 10^{-124}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	27.9
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.48045622450914 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	28.1
Cost	848

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	27.9
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	27.9
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	27.9
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 5.622527198255715 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2858646756872904 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	9.6
Cost	712

\[\begin{array}{l} t_1 := x - \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -8.159561898011284 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3233900258424986 \cdot 10^{-64}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	9.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8.159561898011284 \cdot 10^{+70}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 3.3233900258424986 \cdot 10^{-64}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 16
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934698876908823 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.975711980427134 \cdot 10^{-219}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	16.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+192}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 18
Error	2.4
Cost	576

\[x + \frac{t - z}{\frac{a}{y}} \]

Alternative 19
Error	2.5
Cost	576

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

Alternative 20
Error	30.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022311 
(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)))