Average Error: 0.2 → 0.0
Time: 6.1s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left({b}^{4} + \left({a}^{4} + \left(\left(2 \cdot {a}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right) \cdot {b}^{2} + 4 \cdot \left(a \cdot \left(a + {a}^{2}\right)\right)\right)\right)\right) + -1 \]
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (+
   (pow b 4.0)
   (+
    (pow a 4.0)
    (+
     (* (+ (* 2.0 (pow a 2.0)) (* 4.0 (+ 1.0 (* a -3.0)))) (pow b 2.0))
     (* 4.0 (* a (+ a (pow a 2.0)))))))
  -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 (pow(b, 4.0) + (pow(a, 4.0) + ((((2.0 * pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0)))) * pow(b, 2.0)) + (4.0 * (a * (a + pow(a, 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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 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) + ((((2.0d0 * (a ** 2.0d0)) + (4.0d0 * (1.0d0 + (a * (-3.0d0))))) * (b ** 2.0d0)) + (4.0d0 * (a * (a + (a ** 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 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
public static double code(double a, double b) {
	return (Math.pow(b, 4.0) + (Math.pow(a, 4.0) + ((((2.0 * Math.pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0)))) * Math.pow(b, 2.0)) + (4.0 * (a * (a + Math.pow(a, 2.0))))))) + -1.0;
}

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0
↓
def code(a, b):
	return (math.pow(b, 4.0) + (math.pow(a, 4.0) + ((((2.0 * math.pow(a, 2.0)) + (4.0 * (1.0 + (a * -3.0)))) * math.pow(b, 2.0)) + (4.0 * (a * (a + math.pow(a, 2.0))))))) + -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 Float64(Float64((b ^ 4.0) + Float64((a ^ 4.0) + Float64(Float64(Float64(Float64(2.0 * (a ^ 2.0)) + Float64(4.0 * Float64(1.0 + Float64(a * -3.0)))) * (b ^ 2.0)) + Float64(4.0 * Float64(a * Float64(a + (a ^ 2.0))))))) + -1.0)
end

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end
↓
function tmp = code(a, b)
	tmp = ((b ^ 4.0) + ((a ^ 4.0) + ((((2.0 * (a ^ 2.0)) + (4.0 * (1.0 + (a * -3.0)))) * (b ^ 2.0)) + (4.0 * (a * (a + (a ^ 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[(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[(N[Power[b, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(N[(N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * N[(a + N[Power[a, 2.0], $MachinePrecision]), $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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({b}^{4} + \left({a}^{4} + \left(\left(2 \cdot {a}^{2} + 4 \cdot \left(1 + a \cdot -3\right)\right) \cdot {b}^{2} + 4 \cdot \left(a \cdot \left(a + {a}^{2}\right)\right)\right)\right)\right) + -1

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  3. Taylor expanded in b around 0 0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022210 
(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))