Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Average Error: 0.2 → 0.2

Time: 8.6s

Precision: binary64

Cost: 6720

Math

\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]

↓

\[x \cdot \mathsf{fma}\left(x, -9, 6\right) \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

↓

(FPCore (x) :precision binary64 (* x (fma x -9.0 6.0)))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

↓

double code(double x) {
	return x * fma(x, -9.0, 6.0);
}

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

↓

function code(x)
	return Float64(x * fma(x, -9.0, 6.0))
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * -9.0 + 6.0), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

\[6 \cdot x - 9 \cdot \left(x \cdot x\right) \]

Derivation

1. Initial program 0.2

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]

2. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -9, 6\right)} \]
   Proof

   [Start] 0.2	\[ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   *-commutative [=>] 0.2	\[ \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]
   sub-neg [=>] 0.2	\[ x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x \cdot 3\right)\right)}\right) \]
   distribute-lft-in [=>] 0.2	\[ x \cdot \color{blue}{\left(3 \cdot 2 + 3 \cdot \left(-x \cdot 3\right)\right)} \]
   +-commutative [=>] 0.2	\[ x \cdot \color{blue}{\left(3 \cdot \left(-x \cdot 3\right) + 3 \cdot 2\right)} \]
   *-commutative [=>] 0.2	\[ x \cdot \left(3 \cdot \left(-\color{blue}{3 \cdot x}\right) + 3 \cdot 2\right) \]
   distribute-lft-neg-in [=>] 0.2	\[ x \cdot \left(3 \cdot \color{blue}{\left(\left(-3\right) \cdot x\right)} + 3 \cdot 2\right) \]
   associate-*r* [=>] 0.2	\[ x \cdot \left(\color{blue}{\left(3 \cdot \left(-3\right)\right) \cdot x} + 3 \cdot 2\right) \]
   *-commutative [<=] 0.2	\[ x \cdot \left(\color{blue}{x \cdot \left(3 \cdot \left(-3\right)\right)} + 3 \cdot 2\right) \]
   fma-def [=>] 0.2	\[ x \cdot \color{blue}{\mathsf{fma}\left(x, 3 \cdot \left(-3\right), 3 \cdot 2\right)} \]
   metadata-eval [=>] 0.2	\[ x \cdot \mathsf{fma}\left(x, 3 \cdot \color{blue}{-3}, 3 \cdot 2\right) \]
   metadata-eval [=>] 0.2	\[ x \cdot \mathsf{fma}\left(x, \color{blue}{-9}, 3 \cdot 2\right) \]
   metadata-eval [=>] 0.2	\[ x \cdot \mathsf{fma}\left(x, -9, \color{blue}{6}\right) \]

3. Final simplification 0.2

   \[\leadsto x \cdot \mathsf{fma}\left(x, -9, 6\right) \]

Alternatives

Alternative 1
Error	1.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.68\right):\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]

Alternative 2
Error	1.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	448

\[x \cdot \left(6 + x \cdot -9\right) \]

Alternative 4
Error	20.8
Cost	192

\[x \cdot 6 \]

Alternative 5
Error	62.1
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))