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

Average Error: 5.9 → 1.4

Time: 11.4s

Precision: binary64

Cost: 6980

Math

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

↓

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

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 (/ (- t z) 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, ((t - z) / 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(t - z) / 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[(t - z), $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

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

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

   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

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

Alternatives

Alternative 1
Error	0.4
Cost	1352

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

Alternative 2
Error	30.5
Cost	1308

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-217}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	30.5
Cost	1308

\[\begin{array}{l} t_1 := z \cdot \frac{-y}{a}\\ t_2 := t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-220}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	30.4
Cost	1308

\[\begin{array}{l} t_1 := t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-246}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{-z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	1.6
Cost	1097

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

Alternative 6
Error	9.8
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+26} \lor \neg \left(z \leq 1.66 \cdot 10^{-12}\right) \land \left(z \leq 5 \cdot 10^{+74} \lor \neg \left(z \leq 3 \cdot 10^{+117}\right)\right):\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7
Error	21.4
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-53} \lor \neg \left(x \leq -3.4 \cdot 10^{-92}\right) \land x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
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 9
Error	16.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-111} \lor \neg \left(x \leq 5.4 \cdot 10^{+29}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 10
Error	28.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	29.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	29.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	2.6
Cost	576

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

Alternative 14
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, 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)))