Average Error: 0.2 → 0.1
Time: 3.9s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, 3 \cdot {b}^{2}\right), {b}^{4} + \mathsf{fma}\left(2, \left(b \cdot b\right) \cdot \left(a \cdot a\right), {a}^{4}\right)\right) + -1 \]
(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 a (- a (* a a)) (* 3.0 (pow b 2.0)))
   (+ (pow b 4.0) (fma 2.0 (* (* b b) (* a a)) (pow a 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(a, (a - (a * a)), (3.0 * pow(b, 2.0))), (pow(b, 4.0) + fma(2.0, ((b * b) * (a * a)), pow(a, 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, Float64(a - Float64(a * a)), Float64(3.0 * (b ^ 2.0))), Float64((b ^ 4.0) + fma(2.0, Float64(Float64(b * b) * Float64(a * a)), (a ^ 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[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + N[(2.0 * N[(N[(b * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[a, 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, a - a \cdot a, 3 \cdot {b}^{2}\right), {b}^{4} + \mathsf{fma}\left(2, \left(b \cdot b\right) \cdot \left(a \cdot a\right), {a}^{4}\right)\right) + -1

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in a around 0 0.0

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

  4. Simplified0.0

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

  5. Taylor expanded in a around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022155 
(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))