Average Error: 0.3 → 0.2

Time: 18.0s

Precision: binary64

Cost: 6848

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

Target

Original	0.3
Target	0.2
Herbie	0.2

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

Derivation

Initial program 0.3
\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
Taylor expanded in x around 0 0.2
\[\leadsto \color{blue}{6 \cdot x + -9 \cdot {x}^{2}} \]
Simplified0.2
\[\leadsto \color{blue}{\mathsf{fma}\left(6, x, -9 \cdot \left(x \cdot x\right)\right)} \]
Proof

Alternatives

Alternative 1
Error	2.0
Cost	584

\[\begin{array}{l} t_0 := \left(-9 \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;6 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	448

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

Alternative 3
Error	21.6
Cost	192

\[6 \cdot x \]

Reproduce

herbie shell --seed 2023010 
(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))