Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2

Average Error: 0.0 → 0

Time: 1.5s

Precision: binary64

Cost: 6720

Math

\[\left(x \cdot x\right) \cdot 2 - 1 \]

↓

\[\mathsf{fma}\left(x, x \cdot 2, -1\right) \]

FPCore

(FPCore (x) :precision binary64 (- (* (* x x) 2.0) 1.0))

↓

(FPCore (x) :precision binary64 (fma x (* x 2.0) -1.0))

C

double code(double x) {
	return ((x * x) * 2.0) - 1.0;
}

↓

double code(double x) {
	return fma(x, (x * 2.0), -1.0);
}

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * 2.0) - 1.0)
end

↓

function code(x)
	return fma(x, Float64(x * 2.0), -1.0)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\left(x \cdot x\right) \cdot 2 - 1 \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1\right)} \]
   Proof
   (fma.f64 x (*.f64 x 2) -1): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 x 2) (Rewrite<= metadata-eval (neg.f64 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (*.f64 x 2)) 1)): 2 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) 2)) 1): 0 points increase in error, 0 points decrease in error

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, x \cdot 2, -1\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	448

\[-1 + 2 \cdot \left(x \cdot x\right) \]

Alternative 2
Error	21.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x)
  :name "Numeric.SpecFunctions:logGammaCorrection from math-functions-0.1.5.2"
  :precision binary64
  (- (* (* x x) 2.0) 1.0))