\[\frac{x \cdot x - 3}{6} \]

↓

\[\frac{\mathsf{fma}\left(x, x, -3\right)}{6} \]

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

↓

(FPCore (x) :precision binary64 (/ (fma x x -3.0) 6.0))

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

↓

double code(double x) {
	return fma(x, x, -3.0) / 6.0;
}

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

↓

function code(x)
	return Float64(fma(x, x, -3.0) / 6.0)
end

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

↓

code[x_] := N[(N[(x * x + -3.0), $MachinePrecision] / 6.0), $MachinePrecision]

\frac{x \cdot x - 3}{6}

↓

\frac{\mathsf{fma}\left(x, x, -3\right)}{6}

Derivation

Initial program 0.1
\[\frac{x \cdot x - 3}{6} \]
Simplified0.1
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, -3\right)}{6}} \]
Proof
(/.f64 (fma.f64 x x -3) 6): 0 points increase in error, 0 points decrease in error
(/.f64 (fma.f64 x x (Rewrite<= metadata-eval (neg.f64 3))) 6): 0 points increase in error, 0 points decrease in error
(/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x x) 3)) 6): 1 points increase in error, 0 points decrease in error
Final simplification0.1
\[\leadsto \frac{\mathsf{fma}\left(x, x, -3\right)}{6} \]

Alternatives

Alternative 1
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{6}{x}}\\ \mathbf{if}\;x \leq -23.703143750732394:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.989449587946897 \cdot 10^{-6}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -23.703143750732394:\\ \;\;\;\;\frac{x \cdot x}{6}\\ \mathbf{elif}\;x \leq 3.989449587946897 \cdot 10^{-6}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.02:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	448

\[\frac{x}{\frac{6}{x}} + -0.5 \]

Alternative 5
Error	0.1
Cost	448

\[\frac{-3 + x \cdot x}{6} \]

Alternative 6
Error	21.2
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))

Error

Derivation

Alternatives

Error

Reproduce