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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Precision: binary64

Cost: 6848

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = 3.083437720980777e-240

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

function code(x)
	return fma(Float64(x * x), -9.0, Float64(x * 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[(N[(x * x), $MachinePrecision] * -9.0 + N[(x * 6.0), $MachinePrecision]), $MachinePrecision]

Target

Original	99.8%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 99.8%

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

2. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   Step-by-step derivation

   [Start] 99.8%	\[ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   *-commutative [=>] 99.8%	\[ \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]
   sub-neg [=>] 99.8%	\[ x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x \cdot 3\right)\right)}\right) \]
   distribute-rgt-in [=>] 99.7%	\[ x \cdot \color{blue}{\left(2 \cdot 3 + \left(-x \cdot 3\right) \cdot 3\right)} \]
   metadata-eval [=>] 99.7%	\[ x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]
   distribute-rgt-neg-in [=>] 99.7%	\[ x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]
   associate-*l* [=>] 99.8%	\[ x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]
   metadata-eval [=>] 99.8%	\[ x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]
   metadata-eval [=>] 99.8%	\[ x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Applied egg-rr 99.9%

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

   [Start] 99.8%	\[ x \cdot \left(6 + x \cdot -9\right) \]
   +-commutative [=>] 99.8%	\[ x \cdot \color{blue}{\left(x \cdot -9 + 6\right)} \]
   distribute-lft-in [=>] 99.8%	\[ \color{blue}{x \cdot \left(x \cdot -9\right) + x \cdot 6} \]
   associate-*r* [=>] 99.9%	\[ \color{blue}{\left(x \cdot x\right) \cdot -9} + x \cdot 6 \]
   *-commutative [<=] 99.9%	\[ \left(x \cdot x\right) \cdot -9 + \color{blue}{6 \cdot x} \]
   metadata-eval [<=] 99.9%	\[ \left(x \cdot x\right) \cdot -9 + \color{blue}{\left(2 \cdot 3\right)} \cdot x \]
   associate-*r* [<=] 99.9%	\[ \left(x \cdot x\right) \cdot -9 + \color{blue}{2 \cdot \left(3 \cdot x\right)} \]
   fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(x \cdot x, -9, 2 \cdot \left(3 \cdot x\right)\right)} \]
   associate-*r* [=>] 99.9%	\[ \mathsf{fma}\left(x \cdot x, -9, \color{blue}{\left(2 \cdot 3\right) \cdot x}\right) \]
   metadata-eval [=>] 99.9%	\[ \mathsf{fma}\left(x \cdot x, -9, \color{blue}{6} \cdot x\right) \]
   *-commutative [=>] 99.9%	\[ \mathsf{fma}\left(x \cdot x, -9, \color{blue}{x \cdot 6}\right) \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6848

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

Alternative 2
Accuracy	97.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -9\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	448

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

Alternative 4
Accuracy	41.3%
Cost	192

\[x \cdot 6 \]

Reproduce

herbie shell --seed 2023243 
(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))