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

Average Accuracy: 99.6% → 99.8%

Time: 7.7s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	99.6%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Initial program 99.6%

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

2. Simplified 99.7%

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

   [Start] 99.6	\[ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   *-commutative [=>] 99.6	\[ \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.6	\[ x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]
   distribute-rgt-in [=>] 99.6	\[ x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]
   metadata-eval [=>] 99.6	\[ x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]
   distribute-lft-neg-in [<=] 99.6	\[ x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]
   distribute-rgt-neg-in [=>] 99.6	\[ x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]
   associate-*l* [=>] 99.7	\[ x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]
   metadata-eval [=>] 99.7	\[ x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]
   metadata-eval [=>] 99.7	\[ x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Applied egg-rr 99.8%

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

   [Start] 99.7	\[ x \cdot \left(6 + x \cdot -9\right) \]
   distribute-lft-in [=>] 99.7	\[ \color{blue}{x \cdot 6 + x \cdot \left(x \cdot -9\right)} \]
   *-commutative [<=] 99.7	\[ x \cdot 6 + \color{blue}{\left(x \cdot -9\right) \cdot x} \]
   fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot -9\right) \cdot x\right)} \]
   *-commutative [=>] 99.8	\[ \mathsf{fma}\left(x, 6, \color{blue}{\left(-9 \cdot x\right)} \cdot x\right) \]
   associate-*l* [=>] 99.8	\[ \mathsf{fma}\left(x, 6, \color{blue}{-9 \cdot \left(x \cdot x\right)}\right) \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	96.8%
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
Accuracy	96.8%
Cost	584

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

Alternative 3
Accuracy	99.7%
Cost	576

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

Alternative 4
Accuracy	99.7%
Cost	448

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

Alternative 5
Accuracy	66.0%
Cost	192

\[x \cdot 6 \]

Alternative 6
Accuracy	2.9%
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2023129 
(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))