Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 46.4s

Precision: binary64

Cost: 21376

Math

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

↓

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

FPCore

(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) (+ (* 2.0 (pow (* a b) 2.0)) (pow a 4.0)))
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

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) + ((2.0 * pow((a * b), 2.0)) + pow(a, 4.0))) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

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) + ((2.0d0 * ((a * b) ** 2.0d0)) + (a ** 4.0d0))) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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) + ((2.0 * Math.pow((a * b), 2.0)) + Math.pow(a, 4.0))) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

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) + ((2.0 * math.pow((a * b), 2.0)) + math.pow(a, 4.0))) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

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(Float64((b ^ 4.0) + Float64(Float64(2.0 * (Float64(a * b) ^ 2.0)) + (a ^ 4.0))) + 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

MATLAB

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) + ((2.0 * ((a * b) ^ 2.0)) + (a ^ 4.0))) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

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[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $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]

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. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   rational_best-simplify-47 [=>] 0.0	\[ \left(\color{blue}{\left({b}^{4} + \left({a}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   rational_best-simplify-3 [<=] 0.0	\[ \left(\left({b}^{4} + \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   exponential-simplify-28 [=>] 0.0	\[ \left(\left({b}^{4} + \left(2 \cdot \color{blue}{{\left(a \cdot b\right)}^{2}} + {a}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	9600

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot \left(1 - 3 \cdot a\right)\right) + \left(a \cdot a\right) \cdot \left(1 + a\right)\\ \left(t_0 \cdot 8 - \left(4 \cdot t_0 - {\left(b \cdot b + a \cdot a\right)}^{2}\right)\right) - 1 \end{array} \]

Alternative 2
Error	0.2
Cost	8320

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

Alternative 3
Error	0.2
Cost	8320

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

Alternative 4
Error	2.5
Cost	8200

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -28:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 29.5:\\ \;\;\;\;{b}^{4} + \left(-1 - \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.9
Cost	8200

\[\begin{array}{l} t_0 := \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \mathbf{if}\;a \leq -8.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;{b}^{4} + \left(-1 - \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	3.9
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b \leq -0.52:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7
Error	3.7
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -14:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.7:\\ \;\;\;\;{b}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	13.2
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -0.49:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9
Error	23.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023099 
(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))