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

Average Error: 5.9 → 0.8

Time: 13.4s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

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 -5e+120)
     (fma y (/ (- t z) a) x)
     (if (<= t_1 5e+167) (- x (/ t_1 a)) (- x (/ (- z t) (/ a y)))))))

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 <= -5e+120) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 5e+167) {
		tmp = x - (t_1 / a);
	} else {
		tmp = x - ((z - t) / (a / y));
	}
	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 <= -5e+120)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 5e+167)
		tmp = Float64(x - Float64(t_1 / a));
	else
		tmp = Float64(x - Float64(Float64(z - t) / Float64(a / y)));
	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, -5e+120], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+167], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z - t), $MachinePrecision] / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.9
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 3 regimes

2. if (*.f64 y (-.f64 z t)) < -5.00000000000000019e120

   1. Initial program 16.3

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

   2. Simplified 2.9

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

      [Start] 16.3	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 16.3	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 16.3	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 16.3	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 1.7	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 1.7	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 2.9	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 2.9	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 2.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 2.9	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 2.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 2.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 2.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 2.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

   if -5.00000000000000019e120 < (*.f64 y (-.f64 z t)) < 4.9999999999999997e167

   1. Initial program 0.4

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

   if 4.9999999999999997e167 < (*.f64 y (-.f64 z t))

   1. Initial program 24.5

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

   2. Simplified 0.8

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

      [Start] 24.5	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 0.8	\[ x - \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   3. Applied egg-rr 0.9

      \[\leadsto x - \color{blue}{\frac{z - t}{\frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 5 \cdot 10^{+167}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	33.0
Cost	1440

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{z}}\\ t_2 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+140}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.08 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-258}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	33.1
Cost	1440

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{z}}\\ t_2 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -7.1 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+142}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.08 \cdot 10^{-20}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-279}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+151}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+236}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;x - \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Error	32.1
Cost	1176

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{-20}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-280}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+159}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \end{array} \]

Alternative 6
Error	34.1
Cost	1114

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-102} \lor \neg \left(z \leq 4.8 \cdot 10^{-279}\right) \land \left(z \leq 1.06 \cdot 10^{-256} \lor \neg \left(z \leq 6.5 \cdot 10^{-76}\right) \land z \leq 10^{-53}\right):\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	13.8
Cost	1108

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{a}{t}}\\ t_2 := x - \frac{y \cdot z}{a}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{+175}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	25.2
Cost	976

\[\begin{array}{l} t_1 := y \cdot \frac{t - z}{a}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	24.9
Cost	976

\[\begin{array}{l} t_1 := y \cdot \frac{t - z}{a}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	18.4
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{a}{t}}\\ t_2 := \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \end{array} \]

Alternative 11
Error	30.4
Cost	849

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-195} \lor \neg \left(a \leq -2.15 \cdot 10^{-268}\right) \land a \leq 5.9 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-259} \lor \neg \left(x \leq -2.95 \cdot 10^{-294}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 13
Error	3.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-202}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 14
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023012 
(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)))