Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H

Average Error: 0.2 → 0.2

Time: 4.0s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

Wolfram

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]

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ \frac{x \cdot x - 3}{6} \]
   fma-neg [=>] 0.2	\[ \frac{\color{blue}{\mathsf{fma}\left(x, x, -3\right)}}{6} \]
   metadata-eval [=>] 0.2	\[ \frac{\mathsf{fma}\left(x, x, \color{blue}{-3}\right)}{6} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]

Alternative 2
Error	1.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 2:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 3
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 2:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	448

\[x \cdot \left(x \cdot 0.16666666666666666\right) + -0.5 \]

Alternative 5
Error	0.2
Cost	448

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

Alternative 6
Error	21.5
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))