Average Error: 0.0 → 0
Time: 1.4s
Precision: binary64
Cost: 6592
\[x \cdot 116 - 16 \]
↓
\[\mathsf{fma}\left(x, 116, -16\right) \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
↓
(FPCore (x) :precision binary64 (fma x 116.0 -16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}
↓
double code(double x) {
	return fma(x, 116.0, -16.0);
}

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end
↓
function code(x)
	return fma(x, 116.0, -16.0)
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]
↓
code[x_] := N[(x * 116.0 + -16.0), $MachinePrecision]

x \cdot 116 - 16
↓
\mathsf{fma}\left(x, 116, -16\right)

Error

Derivation

  1. Initial program 0.0

    \[x \cdot 116 - 16 \]

  2. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 116, -16\right)} \]
    Proof
    (fma.f64 x 116 -16): 0 points increase in error, 0 points decrease in error
    (fma.f64 x 116 (Rewrite<= metadata-eval (neg.f64 16))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x 116) 16)): 2 points increase in error, 0 points decrease in error

  3. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, 116, -16\right) \]

Alternatives

Alternative 1
Error1.3
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -0.135:\\ \;\;\;\;x \cdot 116\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;-16\\ \mathbf{else}:\\ \;\;\;\;x \cdot 116\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[-16 + x \cdot 116 \]
Alternative 3
Error31.2
Cost64
\[-16 \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "Data.Colour.CIE:lightness from colour-2.3.3"
  :precision binary64
  (- (* x 116.0) 16.0))