Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 6.7s

Precision: binary64

Cost: 20480

Math

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

↓

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

FPCore

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

C

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

Java

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

Python

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

Julia

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(Float64((b ^ 4.0) + Float64(Float64(2.0 * (Float64(b * a) ^ 2.0)) + (a ^ 4.0))) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

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) + ((2.0 * ((b * a) ^ 2.0)) + (a ^ 4.0))) + (4.0 * (b * b))) - 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[(b * b), $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[(b * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   rational.json-simplify-41 [<=] 0.0	\[ \left(\color{blue}{\left({b}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   exponential.json-simplify-27 [=>] 0.0	\[ \left(\left({b}^{4} + \left(2 \cdot \color{blue}{{\left(a \cdot b\right)}^{2}} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   rational.json-simplify-2 [=>] 0.0	\[ \left(\left({b}^{4} + \left(2 \cdot {\color{blue}{\left(b \cdot a\right)}}^{2} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	7424

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

Alternative 2
Error	2.5
Cost	7304

\[\begin{array}{l} t_0 := {b}^{4} - 1\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 38:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7304

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -0.000135:\\ \;\;\;\;\left({a}^{4} + t_0\right) - 1\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;\left({b}^{4} + t_0\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} - 1\\ \end{array} \]

Alternative 4
Error	2.3
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -0.000135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;{b}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.4
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.98:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6
Error	12.0
Cost	6656

\[{a}^{4} - 1 \]

Alternative 7
Error	23.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023068 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))