Average Error: 0.2 → 0.0
Time: 4.8s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\left(0.5 \cdot {a}^{4} + \mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, \mathsf{fma}\left(4, a \cdot a - {a}^{3}, \mathsf{fma}\left(-0.5, {a}^{4}, {a}^{4} + {b}^{4}\right)\right)\right)\right) + -1 \]
(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
 (+
  (+
   (* 0.5 (pow a 4.0))
   (fma
    (fma 2.0 (* a a) (+ 12.0 (* a 4.0)))
    (* b b)
    (fma
     4.0
     (- (* a a) (pow a 3.0))
     (fma -0.5 (pow a 4.0) (+ (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 ((0.5 * pow(a, 4.0)) + fma(fma(2.0, (a * a), (12.0 + (a * 4.0))), (b * b), fma(4.0, ((a * a) - pow(a, 3.0)), fma(-0.5, pow(a, 4.0), (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(Float64(Float64(0.5 * (a ^ 4.0)) + fma(fma(2.0, Float64(a * a), Float64(12.0 + Float64(a * 4.0))), Float64(b * b), fma(4.0, Float64(Float64(a * a) - (a ^ 3.0)), fma(-0.5, (a ^ 4.0), 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[(N[(0.5 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(a * a), $MachinePrecision] + N[(12.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(4.0 * N[(N[(a * a), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[a, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $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
↓
\left(0.5 \cdot {a}^{4} + \mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, \mathsf{fma}\left(4, a \cdot a - {a}^{3}, \mathsf{fma}\left(-0.5, {a}^{4}, {a}^{4} + {b}^{4}\right)\right)\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. Simplified0.0

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

  3. Taylor expanded in b around inf 0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022180 
(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))