Average Error: 0.2 → 0.0
Time: 2.0s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
↓
\[\left({b}^{4} + \left({a}^{4} + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)\right) + -1 \]
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (+ (pow b 4.0) (+ (pow a 4.0) (* (+ 4.0 (* 2.0 (pow a 2.0))) (pow b 2.0))))
  -1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}
↓
double code(double a, double b) {
	return (pow(b, 4.0) + (pow(a, 4.0) + ((4.0 + (2.0 * pow(a, 2.0))) * pow(b, 2.0)))) + -1.0;
}

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function
↓
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((b ** 4.0d0) + ((a ** 4.0d0) + ((4.0d0 + (2.0d0 * (a ** 2.0d0))) * (b ** 2.0d0)))) + (-1.0d0)
end function

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}
↓
public static double code(double a, double b) {
	return (Math.pow(b, 4.0) + (Math.pow(a, 4.0) + ((4.0 + (2.0 * Math.pow(a, 2.0))) * Math.pow(b, 2.0)))) + -1.0;
}

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0
↓
def code(a, b):
	return (math.pow(b, 4.0) + (math.pow(a, 4.0) + ((4.0 + (2.0 * math.pow(a, 2.0))) * math.pow(b, 2.0)))) + -1.0

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end
↓
function code(a, b)
	return Float64(Float64((b ^ 4.0) + Float64((a ^ 4.0) + Float64(Float64(4.0 + Float64(2.0 * (a ^ 2.0))) * (b ^ 2.0)))) + -1.0)
end

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end
↓
function tmp = code(a, b)
	tmp = ((b ^ 4.0) + ((a ^ 4.0) + ((4.0 + (2.0 * (a ^ 2.0))) * (b ^ 2.0)))) + -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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[a_, b_] := N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(4.0 + N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  3. Taylor expanded in a around inf 0.1

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

  4. Taylor expanded in b around 0 0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022197 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))