Average Error: 0.1 → 0.2
Time: 2.2s
Precision: binary64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
↓
\[1 - \mathsf{fma}\left(0.12, {x}^{2}, x \cdot 0.253\right) \]
(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))
↓
(FPCore (x) :precision binary64 (- 1.0 (fma 0.12 (pow x 2.0) (* x 0.253))))
double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}
↓
double code(double x) {
	return 1.0 - fma(0.12, pow(x, 2.0), (x * 0.253));
}

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end
↓
function code(x)
	return Float64(1.0 - fma(0.12, (x ^ 2.0), Float64(x * 0.253)))
end

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(1.0 - N[(0.12 * N[Power[x, 2.0], $MachinePrecision] + N[(x * 0.253), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
1 - \mathsf{fma}\left(0.12, {x}^{2}, x \cdot 0.253\right)

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]

  3. Taylor expanded in x around 0 0.2

    \[\leadsto \color{blue}{1 - \left(0.12 \cdot {x}^{2} + 0.253 \cdot x\right)} \]

  4. Applied fma-def_binary640.2

    \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(0.12, {x}^{2}, 0.253 \cdot x\right)} \]

  5. Final simplification0.2

    \[\leadsto 1 - \mathsf{fma}\left(0.12, {x}^{2}, x \cdot 0.253\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))