?

Average Error: 0.31% → 0.03%
Time: 35.0s
Precision: binary64

?

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left({a}^{2}, 1 - a, \left(3 + a\right) \cdot {b}^{2}\right), 2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) - 1 \]
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (fma
   4.0
   (fma (pow a 2.0) (- 1.0 a) (* (+ 3.0 a) (pow b 2.0)))
   (+ (* 2.0 (* (pow b 2.0) (pow a 2.0))) (+ (pow a 4.0) (pow b 4.0))))
  1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}
↓
double code(double a, double b) {
	return fma(4.0, fma(pow(a, 2.0), (1.0 - a), ((3.0 + a) * pow(b, 2.0))), ((2.0 * (pow(b, 2.0) * pow(a, 2.0))) + (pow(a, 4.0) + pow(b, 4.0)))) - 1.0;
}

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end
↓
function code(a, b)
	return Float64(fma(4.0, fma((a ^ 2.0), Float64(1.0 - a), Float64(Float64(3.0 + a) * (b ^ 2.0))), Float64(Float64(2.0 * Float64((b ^ 2.0) * (a ^ 2.0))) + Float64((a ^ 4.0) + (b ^ 4.0)))) - 1.0)
end

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[a_, b_] := N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + N[(N[(3.0 + a), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left({a}^{2}, 1 - a, \left(3 + a\right) \cdot {b}^{2}\right), 2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) - 1

Error?

Derivation?

  1. Initial program 0.31

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

  2. Simplified0.31

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left({a}^{2}, 1 - a, \left(3 + a\right) \cdot {b}^{2}\right), {\left(\mathsf{fma}\left(a, a, {b}^{2}\right)\right)}^{2}\right) - 1} \]
    Proof

  3. Taylor expanded in b around 0 0.17

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left({a}^{2}, 1 - a, \left(3 + a\right) \cdot {b}^{2}\right), \color{blue}{2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left({\left({a}^{2}\right)}^{2} + {b}^{4}\right)}\right) - 1 \]

  4. Taylor expanded in a around 0 0.03

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left({a}^{2}, 1 - a, \left(3 + a\right) \cdot {b}^{2}\right), 2 \cdot \left({b}^{2} \cdot {a}^{2}\right) + \left(\color{blue}{{a}^{4}} + {b}^{4}\right)\right) - 1 \]

Reproduce?

herbie shell --seed 2023136 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))