Average Error: 0.2 → 0.0
Time: 4.0s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left({a}^{4}, 0.5, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, -{a}^{3}\right), \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(4, a, 12\right)\right), b \cdot b, -1\right)\right) \]
(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
  (pow a 4.0)
  0.5
  (+
   (fma 4.0 (fma a a (- (pow a 3.0))) (fma (pow a 4.0) 0.5 (pow b 4.0)))
   (fma (fma a (* a 2.0) (fma 4.0 a 12.0)) (* b b) -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(pow(a, 4.0), 0.5, (fma(4.0, fma(a, a, -pow(a, 3.0)), fma(pow(a, 4.0), 0.5, pow(b, 4.0))) + fma(fma(a, (a * 2.0), fma(4.0, a, 12.0)), (b * b), -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 fma((a ^ 4.0), 0.5, Float64(fma(4.0, fma(a, a, Float64(-(a ^ 3.0))), fma((a ^ 4.0), 0.5, (b ^ 4.0))) + fma(fma(a, Float64(a * 2.0), fma(4.0, a, 12.0)), Float64(b * b), -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[Power[a, 4.0], $MachinePrecision] * 0.5 + N[(N[(4.0 * N[(a * a + (-N[Power[a, 3.0], $MachinePrecision])), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] * 0.5 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(a * 2.0), $MachinePrecision] + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $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({a}^{4}, 0.5, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, -{a}^{3}\right), \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(4, a, 12\right)\right), b \cdot b, -1\right)\right)

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. Simplified0.0

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

  3. Applied egg-rr0.0

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

  4. Taylor expanded in b around inf 0.0

    \[\leadsto \color{blue}{\left(0.5 \cdot {a}^{4} + \left(\left(2 \cdot {a}^{2} + 4 \cdot \left(3 + a\right)\right) \cdot {b}^{2} + \left(4 \cdot \left({a}^{2} - {a}^{3}\right) + \left(-0.5 \cdot {a}^{4} + \left({a}^{4} + {b}^{4}\right)\right)\right)\right)\right) - 1} \]

  5. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{4}, 0.5, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, -{a}^{3}\right), \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(4, a, 12\right)\right), b \cdot b, -1\right)\right)} \]

  6. Final simplification0.0

    \[\leadsto \mathsf{fma}\left({a}^{4}, 0.5, \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, -{a}^{3}\right), \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(4, a, 12\right)\right), b \cdot b, -1\right)\right) \]

Reproduce

herbie shell --seed 2022178 
(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))