Average Error: 0.2 → 0.0
Time: 3.7s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left({a}^{4}, 0.5, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a, a\right), 4, \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), 4\right), -1\right)\right) \]
(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
 (fma
  (pow a 4.0)
  0.5
  (+
   (fma (* a (fma a a a)) 4.0 (fma (pow a 4.0) 0.5 (pow b 4.0)))
   (fma (* b b) (fma a (fma a 2.0 -12.0) 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) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return fma(pow(a, 4.0), 0.5, (fma((a * fma(a, a, a)), 4.0, fma(pow(a, 4.0), 0.5, pow(b, 4.0))) + fma((b * b), fma(a, fma(a, 2.0, -12.0), 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(1.0 - Float64(3.0 * a)))))) - 1.0)
end
↓
function code(a, b)
	return fma((a ^ 4.0), 0.5, Float64(fma(Float64(a * fma(a, a, a)), 4.0, fma((a ^ 4.0), 0.5, (b ^ 4.0))) + fma(Float64(b * b), fma(a, fma(a, 2.0, -12.0), 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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[a_, b_] := N[(N[Power[a, 4.0], $MachinePrecision] * 0.5 + N[(N[(N[(a * N[(a * a + a), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(N[Power[a, 4.0], $MachinePrecision] * 0.5 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * N[(a * 2.0 + -12.0), $MachinePrecision] + 4.0), $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
↓
\mathsf{fma}\left({a}^{4}, 0.5, \mathsf{fma}\left(a \cdot \mathsf{fma}\left(a, a, a\right), 4, \mathsf{fma}\left({a}^{4}, 0.5, {b}^{4}\right)\right) + \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 2, -12\right), 4\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. Taylor expanded in b around inf 0.1

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

  4. Simplified0.0

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

  5. Applied egg-rr0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022169 
(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))