Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 6.0s

Precision: binary64

Cost: 27712

Math

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

↓

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

FPCore

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

↓

double code(double a, double b) {
	return (4.0 * ((a * (a - (a * a))) + ((b * b) * (a + 3.0)))) + (((2.0 * (pow(b, 2.0) * pow(a, 2.0))) + (pow(b, 4.0) + pow(a, 4.0))) + -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) * (3.0d0 + a))))) - 1.0d0
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (4.0d0 * ((a * (a - (a * a))) + ((b * b) * (a + 3.0d0)))) + (((2.0d0 * ((b ** 2.0d0) * (a ** 2.0d0))) + ((b ** 4.0d0) + (a ** 4.0d0))) + (-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) * (3.0 + a))))) - 1.0;
}

↓

public static double code(double a, double b) {
	return (4.0 * ((a * (a - (a * a))) + ((b * b) * (a + 3.0)))) + (((2.0 * (Math.pow(b, 2.0) * Math.pow(a, 2.0))) + (Math.pow(b, 4.0) + Math.pow(a, 4.0))) + -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) * (3.0 + a))))) - 1.0

↓

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

↓

function code(a, b)
	return Float64(Float64(4.0 * Float64(Float64(a * Float64(a - Float64(a * a))) + Float64(Float64(b * b) * Float64(a + 3.0)))) + Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) * (a ^ 2.0))) + Float64((b ^ 4.0) + (a ^ 4.0))) + -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) * (3.0 + a))))) - 1.0;
end

↓

function tmp = code(a, b)
	tmp = (4.0 * ((a * (a - (a * a))) + ((b * b) * (a + 3.0)))) + (((2.0 * ((b ^ 2.0) * (a ^ 2.0))) + ((b ^ 4.0) + (a ^ 4.0))) + -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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := N[(N[(4.0 * N[(N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   rational_best-simplify-25 [=>] 0.2	\[ \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) + -1} \]
   rational_best-simplify-1 [=>] 0.2	\[ \color{blue}{-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right)} \]
   rational_best-simplify-1 [=>] 0.2	\[ -1 + \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right)} \]
   rational_best-simplify-115 [=>] 0.2	\[ \color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) + \left({\left(a \cdot a + b \cdot b\right)}^{2} + -1\right)} \]

3. Taylor expanded in a around 0 0.0

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

4. Simplified 0.0

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

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	8192

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

Alternative 2
Error	1.7
Cost	8072

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(a + 3\right)\\ t_1 := 4 \cdot \left(a \cdot \left(a - a \cdot a\right) + t_0\right) + \left({a}^{4} + -1\right)\\ \mathbf{if}\;a \leq -0.0005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-25}:\\ \;\;\;\;4 \cdot \left(a \cdot a + t_0\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.0
Cost	7816

\[\begin{array}{l} \mathbf{if}\;b \leq -31.5:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 130:\\ \;\;\;\;4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + \left({a}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 4
Error	2.5
Cost	7816

\[\begin{array}{l} \mathbf{if}\;a \leq -4:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7.9:\\ \;\;\;\;4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5
Error	12.1
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b \leq -0.285:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 0.3:\\ \;\;\;\;4 \cdot {a}^{2} - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6
Error	42.5
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -1.82:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.5:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7
Error	51.8
Cost	6528

\[{a}^{4} \]

Reproduce

herbie shell --seed 2023104 
(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))