Average Error: 62.0 → 0
Time: 2.6s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\\ \mathsf{fma}\left(9, {x}^{4}, -t_0\right) + \mathsf{fma}\left(-\mathsf{fma}\left(y, y, -2\right), y \cdot y, t_0\right) \end{array} \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y y -2.0) (* y y))))
   (+ (fma 9.0 (pow x 4.0) (- t_0)) (fma (- (fma y y -2.0)) (* y y) t_0))))
double code(double x, double y) {
	return (9.0 * pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}
↓
double code(double x, double y) {
	double t_0 = fma(y, y, -2.0) * (y * y);
	return fma(9.0, pow(x, 4.0), -t_0) + fma(-fma(y, y, -2.0), (y * y), t_0);
}

function code(x, y)
	return Float64(Float64(9.0 * (x ^ 4.0)) - Float64(Float64(y * y) * Float64(Float64(y * y) - 2.0)))
end
↓
function code(x, y)
	t_0 = Float64(fma(y, y, -2.0) * Float64(y * y))
	return Float64(fma(9.0, (x ^ 4.0), Float64(-t_0)) + fma(Float64(-fma(y, y, -2.0)), Float64(y * y), t_0))
end

code[x_, y_] := N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := Block[{t$95$0 = N[(N[(y * y + -2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-N[(y * y + -2.0), $MachinePrecision]) * N[(y * y), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\\
\mathsf{fma}\left(9, {x}^{4}, -t_0\right) + \mathsf{fma}\left(-\mathsf{fma}\left(y, y, -2\right), y \cdot y, t_0\right)
\end{array}

Error

Derivation

  1. Initial program 62.0

    \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]

  2. Simplified62.0

    \[\leadsto \color{blue}{9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y, -2\right)} \]

  3. Applied prod-diff_binary640

    \[\leadsto \color{blue}{\mathsf{fma}\left(9, {x}^{4}, -\mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\right) + \mathsf{fma}\left(-\mathsf{fma}\left(y, y, -2\right), y \cdot y, \mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\right)} \]

  4. Final simplification0

    \[\leadsto \mathsf{fma}\left(9, {x}^{4}, -\mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\right) + \mathsf{fma}\left(-\mathsf{fma}\left(y, y, -2\right), y \cdot y, \mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))