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

Average Error: 6.0 → 0.8

Time: 10.6s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.6e-73) (not (<= a 5e-17)))
   (fma y (/ (- t z) a) x)
   (- x (/ (* y (- z t)) 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 ((a <= -7.6e-73) || !(a <= 5e-17)) {
		tmp = fma(y, ((t - z) / a), x);
	} else {
		tmp = x - ((y * (z - t)) / 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 ((a <= -7.6e-73) || !(a <= 5e-17))
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	else
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / 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[a, -7.6e-73], N[Not[LessEqual[a, 5e-17]], $MachinePrecision]], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.0
Target	0.6
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 a < -7.6000000000000005e-73 or 4.9999999999999999e-17 < a

   1. Initial program 8.3

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

   2. Simplified 0.8

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

      [Start] 8.3	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 8.3	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 8.3	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 8.3	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 1.9	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 1.9	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 0.8	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 0.8	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 0.8	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 0.8	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 0.8	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 0.8	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 0.8	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 0.8	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

   if -7.6000000000000005e-73 < a < 4.9999999999999999e-17

   1. Initial program 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-73} \lor \neg \left(a \leq 5 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.3
Cost	1241

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ t_2 := y \cdot \frac{t - z}{a}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+58} \lor \neg \left(y \leq 6 \cdot 10^{+94}\right) \land y \leq 8.8 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.4
Cost	1241

\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ t_2 := y \cdot \frac{t - z}{a}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+58}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+96} \lor \neg \left(y \leq 1.22 \cdot 10^{+210}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	1.4
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+58} \lor \neg \left(z - t \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 4
Error	30.0
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -185000000000:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	22.1
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{+19} \lor \neg \left(x \leq -6.8 \cdot 10^{-134}\right) \land x \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	19.2
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3900000000 \lor \neg \left(x \leq -4.7 \cdot 10^{-38}\right) \land x \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.9
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -480000000000 \lor \neg \left(x \leq -1.75 \cdot 10^{-135}\right) \land x \leq 4.5 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	29.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1300000000000:\\ \;\;\;\;z \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-133}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	0.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-73} \lor \neg \left(a \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternative 10
Error	10.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-16} \lor \neg \left(t \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 11
Error	2.5
Cost	576

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

Alternative 12
Error	30.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(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)))