Average Error: 0.2 → 0.0
Time: 17.6s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{a \cdot a}\\ \left(0.5 \cdot {a}^{4} + \left(\left(2 \cdot {a}^{2} + 4 \cdot \left(a + 3\right)\right) \cdot {b}^{2} + \left(4 \cdot \left(\mathsf{fma}\left(t_0 \cdot t_0, t_0, -{a}^{3}\right) + \mathsf{fma}\left(a \cdot \left(-a\right), a, {a}^{3}\right)\right) + \left({a}^{4} \cdot -0.5 + \left({a}^{4} + {b}^{4}\right)\right)\right)\right)\right) + -1 \end{array} \]
(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
 (let* ((t_0 (cbrt (* a a))))
   (+
    (+
     (* 0.5 (pow a 4.0))
     (+
      (* (+ (* 2.0 (pow a 2.0)) (* 4.0 (+ a 3.0))) (pow b 2.0))
      (+
       (*
        4.0
        (+
         (fma (* t_0 t_0) t_0 (- (pow a 3.0)))
         (fma (* a (- a)) a (pow a 3.0))))
       (+ (* (pow a 4.0) -0.5) (+ (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) {
	double t_0 = cbrt((a * a));
	return ((0.5 * pow(a, 4.0)) + ((((2.0 * pow(a, 2.0)) + (4.0 * (a + 3.0))) * pow(b, 2.0)) + ((4.0 * (fma((t_0 * t_0), t_0, -pow(a, 3.0)) + fma((a * -a), a, pow(a, 3.0)))) + ((pow(a, 4.0) * -0.5) + (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)
	t_0 = cbrt(Float64(a * a))
	return Float64(Float64(Float64(0.5 * (a ^ 4.0)) + Float64(Float64(Float64(Float64(2.0 * (a ^ 2.0)) + Float64(4.0 * Float64(a + 3.0))) * (b ^ 2.0)) + Float64(Float64(4.0 * Float64(fma(Float64(t_0 * t_0), t_0, Float64(-(a ^ 3.0))) + fma(Float64(a * Float64(-a)), a, (a ^ 3.0)))) + Float64(Float64((a ^ 4.0) * -0.5) + 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_] := Block[{t$95$0 = N[Power[N[(a * a), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[(0.5 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0 + (-N[Power[a, 3.0], $MachinePrecision])), $MachinePrecision] + N[(N[(a * (-a)), $MachinePrecision] * a + N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 4.0], $MachinePrecision] * -0.5), $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
↓
\begin{array}{l}
t_0 := \sqrt[3]{a \cdot a}\\
\left(0.5 \cdot {a}^{4} + \left(\left(2 \cdot {a}^{2} + 4 \cdot \left(a + 3\right)\right) \cdot {b}^{2} + \left(4 \cdot \left(\mathsf{fma}\left(t_0 \cdot t_0, t_0, -{a}^{3}\right) + \mathsf{fma}\left(a \cdot \left(-a\right), a, {a}^{3}\right)\right) + \left({a}^{4} \cdot -0.5 + \left({a}^{4} + {b}^{4}\right)\right)\right)\right)\right) + -1
\end{array}

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

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022325 
(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))