Average Error: 0.2 → 0.0
Time: 3.3s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[4 \cdot \mathsf{fma}\left(a, a, {a}^{3}\right) + \left(\left(\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) - \mathsf{fma}\left(12, a \cdot \left(b \cdot b\right), 1\right)\right) \]
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (* 4.0 (fma a a (pow a 3.0)))
  (-
   (+ (* (* b b) (+ 4.0 (* 2.0 (* a a)))) (+ (pow b 4.0) (pow a 4.0)))
   (fma 12.0 (* a (* 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) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return (4.0 * fma(a, a, pow(a, 3.0))) + ((((b * b) * (4.0 + (2.0 * (a * a)))) + (pow(b, 4.0) + pow(a, 4.0))) - fma(12.0, (a * (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(1.0 - Float64(3.0 * a)))))) - 1.0)
end
↓
function code(a, b)
	return Float64(Float64(4.0 * fma(a, a, (a ^ 3.0))) + Float64(Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(2.0 * Float64(a * a)))) + Float64((b ^ 4.0) + (a ^ 4.0))) - fma(12.0, Float64(a * 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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[a_, b_] := N[(N[(4.0 * N[(a * a + N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(12.0 * N[(a * N[(b * b), $MachinePrecision]), $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(1 - 3 \cdot a\right)\right)\right) - 1
↓
4 \cdot \mathsf{fma}\left(a, a, {a}^{3}\right) + \left(\left(\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) - \mathsf{fma}\left(12, a \cdot \left(b \cdot b\right), 1\right)\right)

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  3. Applied egg-rr15.7

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

  4. Taylor expanded in b around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022162 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))