Average Error: 0.2 → 0.0
Time: 5.3s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\left(\mathsf{fma}\left(a \cdot a - {a}^{3}, 4, {a}^{4}\right) + \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(a, 4, 12\right)\right)\right) + \left(-1 + {b}^{4}\right) \]
(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 (- (* a a) (pow a 3.0)) 4.0 (pow a 4.0))
   (* (* b b) (fma a (* a 2.0) (fma a 4.0 12.0))))
  (+ -1.0 (pow b 4.0))))
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(((a * a) - pow(a, 3.0)), 4.0, pow(a, 4.0)) + ((b * b) * fma(a, (a * 2.0), fma(a, 4.0, 12.0)))) + (-1.0 + pow(b, 4.0));
}

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(Float64(fma(Float64(Float64(a * a) - (a ^ 3.0)), 4.0, (a ^ 4.0)) + Float64(Float64(b * b) * fma(a, Float64(a * 2.0), fma(a, 4.0, 12.0)))) + Float64(-1.0 + (b ^ 4.0)))
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[(N[(N[(N[(a * a), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * 4.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * N[(a * 2.0), $MachinePrecision] + N[(a * 4.0 + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[Power[b, 4.0], $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
↓
\left(\mathsf{fma}\left(a \cdot a - {a}^{3}, 4, {a}^{4}\right) + \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, a \cdot 2, \mathsf{fma}\left(a, 4, 12\right)\right)\right) + \left(-1 + {b}^{4}\right)

Error

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. Taylor expanded in b around 0 0.0

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

  4. Simplified0.0

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

  5. Applied egg-rr0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022211 
(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))