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

Average Error: 6.2 → 0.6

Time: 10.7s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00019 \lor \neg \left(y \leq 2.02 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -0.00019) (not (<= y 2.02e-16)))
   (fma y (/ (- t z) a) x)
   (+ x (/ (* y (- t z)) a))))

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 ((y <= -0.00019) || !(y <= 2.02e-16)) {
		tmp = fma(y, ((t - z) / a), x);
	} else {
		tmp = x + ((y * (t - z)) / a);
	}
	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 ((y <= -0.00019) || !(y <= 2.02e-16))
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	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[Or[LessEqual[y, -0.00019], N[Not[LessEqual[y, 2.02e-16]], $MachinePrecision]], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.2
Target	0.7
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if y < -1.9000000000000001e-4 or 2.02000000000000004e-16 < y

   1. Initial program 14.6

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

   2. Simplified 0.9

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

      [Start] 14.6	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 14.6	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 14.6	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 14.6	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 2.7	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 2.7	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 0.9	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 0.9	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 0.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 0.9	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 0.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 0.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 0.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 0.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

   if -1.9000000000000001e-4 < y < 2.02000000000000004e-16

   1. Initial program 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00019 \lor \neg \left(y \leq 2.02 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+176}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - z}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Error	19.9
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -470000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	16.7
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(t - z\right)\\ t_2 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -118000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	16.1
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-66}:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-220}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	20.8
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+20} \lor \neg \left(x \leq -3.5 \cdot 10^{-62}\right) \land x \leq 2.8 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	27.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	27.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-291}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-92}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-90}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \end{array} \]

Alternative 10
Error	27.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	27.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-87}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.5
Cost	576

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

Alternative 13
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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)))