From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%

Time: 2.2s

Precision: binary64

Cost: 19904

\[x = 10864 \land y = 18817\]

Math

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

↓

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

FPCore

(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))

↓

(FPCore (x y)
 :precision binary64
 (fma (fma y y -2.0) (- (* y y)) (* 9.0 (pow x 4.0))))

C

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) {
	return fma(fma(y, y, -2.0), -(y * y), (9.0 * pow(x, 4.0)));
}

Julia

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)
	return fma(fma(y, y, -2.0), Float64(-Float64(y * y)), Float64(9.0 * (x ^ 4.0)))
end

Wolfram

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_] := N[(N[(y * y + -2.0), $MachinePrecision] * (-N[(y * y), $MachinePrecision]) + N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

2. Applied egg-rr 100.0%

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

   [Start] 3.1%	\[ 9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
   sub-neg [=>] 3.1%	\[ \color{blue}{9 \cdot {x}^{4} + \left(-\left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\right)} \]
   +-commutative [=>] 3.1%	\[ \color{blue}{\left(-\left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\right) + 9 \cdot {x}^{4}} \]
   *-commutative [=>] 3.1%	\[ \left(-\color{blue}{\left(y \cdot y - 2\right) \cdot \left(y \cdot y\right)}\right) + 9 \cdot {x}^{4} \]
   distribute-rgt-neg-in [=>] 3.1%	\[ \color{blue}{\left(y \cdot y - 2\right) \cdot \left(-y \cdot y\right)} + 9 \cdot {x}^{4} \]
   fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(y \cdot y - 2, -y \cdot y, 9 \cdot {x}^{4}\right)} \]
   fma-neg [=>] 100.0%	\[ \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y, -2\right)}, -y \cdot y, 9 \cdot {x}^{4}\right) \]
   metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\mathsf{fma}\left(y, y, \color{blue}{-2}\right), -y \cdot y, 9 \cdot {x}^{4}\right) \]
   distribute-rgt-neg-in [=>] 100.0%	\[ \mathsf{fma}\left(\mathsf{fma}\left(y, y, -2\right), \color{blue}{y \cdot \left(-y\right)}, 9 \cdot {x}^{4}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	19904

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

Alternative 2
Accuracy	9.6%
Cost	6656

\[9 \cdot {x}^{4} \]

Alternative 3
Accuracy	1.5%
Cost	6592

\[-{y}^{4} \]

Reproduce

herbie shell --seed 2023178 
(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))))