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

Average Error: 0.23% → 0.14%

Time: 10.4s

Precision: binary64

Cost: 13120

Math

\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

↓

\[\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right) \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

↓

(FPCore (x) :precision binary64 (fma x (fma x 9.0 -12.0) 3.0))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

↓

double code(double x) {
	return fma(x, fma(x, 9.0, -12.0), 3.0);
}

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

↓

function code(x)
	return fma(x, fma(x, 9.0, -12.0), 3.0)
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * 9.0 + -12.0), $MachinePrecision] + 3.0), $MachinePrecision]

Target

Original	0.23%
Target	0.14%
Herbie	0.14%

\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \]

Derivation

1. Initial program 0.23

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

2. Simplified 0.14

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]
   Proof

   [Start] 0.23	\[ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   distribute-rgt-in [=>] 0.21	\[ \color{blue}{\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) \cdot 3 + 1 \cdot 3} \]
   *-commutative [=>] 0.21	\[ \left(\color{blue}{x \cdot \left(x \cdot 3\right)} - x \cdot 4\right) \cdot 3 + 1 \cdot 3 \]
   distribute-lft-out-- [=>] 0.2	\[ \color{blue}{\left(x \cdot \left(x \cdot 3 - 4\right)\right)} \cdot 3 + 1 \cdot 3 \]
   associate-*l* [=>] 0.21	\[ \color{blue}{x \cdot \left(\left(x \cdot 3 - 4\right) \cdot 3\right)} + 1 \cdot 3 \]
   metadata-eval [=>] 0.21	\[ x \cdot \left(\left(x \cdot 3 - 4\right) \cdot 3\right) + \color{blue}{3} \]
   fma-def [=>] 0.21	\[ \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 3 - 4\right) \cdot 3, 3\right)} \]
   *-commutative [=>] 0.21	\[ \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3 - 4\right)}, 3\right) \]
   sub-neg [=>] 0.21	\[ \mathsf{fma}\left(x, 3 \cdot \color{blue}{\left(x \cdot 3 + \left(-4\right)\right)}, 3\right) \]
   distribute-lft-in [=>] 0.21	\[ \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3\right) + 3 \cdot \left(-4\right)}, 3\right) \]
   *-commutative [<=] 0.21	\[ \mathsf{fma}\left(x, \color{blue}{\left(x \cdot 3\right) \cdot 3} + 3 \cdot \left(-4\right), 3\right) \]
   associate-*l* [=>] 0.14	\[ \mathsf{fma}\left(x, \color{blue}{x \cdot \left(3 \cdot 3\right)} + 3 \cdot \left(-4\right), 3\right) \]
   fma-def [=>] 0.14	\[ \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 3 \cdot 3, 3 \cdot \left(-4\right)\right)}, 3\right) \]
   metadata-eval [=>] 0.14	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{9}, 3 \cdot \left(-4\right)\right), 3\right) \]
   metadata-eval [=>] 0.14	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, 3 \cdot \color{blue}{-4}\right), 3\right) \]
   metadata-eval [=>] 0.14	\[ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, \color{blue}{-12}\right), 3\right) \]

3. Final simplification 0.14

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right) \]

Alternatives

Alternative 1
Error	1.46%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;x \cdot \left(-12 + x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3 + x \cdot -12\\ \end{array} \]

Alternative 2
Error	0.14%
Cost	704

\[\left(3 + x \cdot -12\right) + 9 \cdot \left(x \cdot x\right) \]

Alternative 3
Error	2.95%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 4
Error	2.95%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 5
Error	2.19%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 6
Error	0.14%
Cost	576

\[3 + x \cdot \left(-12 + x \cdot 9\right) \]

Alternative 7
Error	32.91%
Cost	64

\[3 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))