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

Average Error: 0.0 → 0.0

Time: 7.3s

Precision: binary64

Cost: 6976

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\left(x + y\right) \cdot \left(1 - z\right) \]

↓

\[\mathsf{fma}\left(1 - z, x, \left(1 - z\right) \cdot y\right) \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (- 1.0 z)))

↓

(FPCore (x y z) :precision binary64 (fma (- 1.0 z) x (* (- 1.0 z) y)))

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

↓

double code(double x, double y, double z) {
	return fma((1.0 - z), x, ((1.0 - z) * y));
}

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(1.0 - z))
end

↓

function code(x, y, z)
	return fma(Float64(1.0 - z), x, Float64(Float64(1.0 - z) * y))
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(1.0 - z), $MachinePrecision] * x + N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\left(x + y\right) \cdot \left(1 - z\right) \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{x + \left(y + \left(x + y\right) \cdot \left(-z\right)\right)} \]

3. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{\left(-1 \cdot z + 1\right) \cdot x + \left(-1 \cdot \left(y \cdot z\right) + y\right)} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, y - y \cdot z\right)} \]
   Proof

   [Start] 0.0	\[ \left(-1 \cdot z + 1\right) \cdot x + \left(-1 \cdot \left(y \cdot z\right) + y\right) \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot z + 1, x, -1 \cdot \left(y \cdot z\right) + y\right)} \]
   fma-def [=>] 0.0	\[ \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1, z, 1\right)}, x, -1 \cdot \left(y \cdot z\right) + y\right) \]
   +-commutative [<=] 0.0	\[ \mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, \color{blue}{y + -1 \cdot \left(y \cdot z\right)}\right) \]
   *-commutative [<=] 0.0	\[ \mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, y + -1 \cdot \color{blue}{\left(z \cdot y\right)}\right) \]
   mul-1-neg [=>] 0.0	\[ \mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, y + \color{blue}{\left(-z \cdot y\right)}\right) \]
   sub-neg [<=] 0.0	\[ \mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, \color{blue}{y - z \cdot y}\right) \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(\mathsf{fma}\left(-1, z, 1\right), x, y - \color{blue}{y \cdot z}\right) \]

5. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - z, x, \left(1 - z\right) \cdot y\right) \cdot 1} \]

6. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(1 - z, x, \left(1 - z\right) \cdot y\right) \]

Alternatives

Alternative 1
Error	12.5
Cost	980

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{+119}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -11.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.3
Cost	980

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+118}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+64}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.3
Cost	980

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+64}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.8
Cost	916

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ t_1 := z \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -19.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.8
Cost	905

\[\begin{array}{l} \mathbf{if}\;1 - z \leq -1 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Error	12.7
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -25.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7
Error	0.0
Cost	448

\[\left(1 - z\right) \cdot \left(x + y\right) \]

Alternative 8
Error	31.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9
Error	23.9
Cost	192

\[x + y \]

Alternative 10
Error	43.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))