Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 8.0s

Precision: binary64

Cost: 20480

Math

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

↓

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

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) (+ (pow a 4.0) (* 2.0 (pow (* b a) 2.0)))) (* 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) + (pow(a, 4.0) + (2.0 * pow((b * a), 2.0)))) + (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) + ((a ** 4.0d0) + (2.0d0 * ((b * a) ** 2.0d0)))) + (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) + (Math.pow(a, 4.0) + (2.0 * Math.pow((b * a), 2.0)))) + (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) + (math.pow(a, 4.0) + (2.0 * math.pow((b * a), 2.0)))) + (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((a ^ 4.0) + Float64(2.0 * (Float64(b * a) ^ 2.0)))) + 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) + ((a ^ 4.0) + (2.0 * ((b * a) ^ 2.0)))) + (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[Power[a, 4.0], $MachinePrecision] + N[(2.0 * N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision]), $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

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

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   +-commutative [=>] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left({b}^{4} + {a}^{4}\right)}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   fma-def [=>] 0.0	\[ \left(\color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4} + {a}^{4}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}, {b}^{4} + {a}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13504

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

Alternative 2
Error	0.1
Cost	7936

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

Alternative 3
Error	2.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -56:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + 2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	7424

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

Alternative 5
Error	1.8
Cost	7305

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -30 \lor \neg \left(a \leq 0.0011\right):\\ \;\;\;\;\left({a}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	1.7
Cost	7305

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -30 \lor \neg \left(a \leq 0.00098\right):\\ \;\;\;\;\left({a}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + t_0\right) + -1\\ \end{array} \]

Alternative 7
Error	1.8
Cost	6921

\[\begin{array}{l} \mathbf{if}\;a \leq -30 \lor \neg \left(a \leq 0.0075\right):\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 8
Error	1.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \leq -30 \lor \neg \left(a \leq 0.00022\right):\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \end{array} \]

Alternative 9
Error	1.9
Cost	1225

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -30 \lor \neg \left(a \leq 0.0019\right):\\ \;\;\;\;\left(t_0 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 10
Error	11.8
Cost	704

\[\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1 \]

Alternative 11
Error	12.7
Cost	576

\[\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1 \]

Alternative 12
Error	22.8
Cost	448

\[b \cdot \left(b \cdot 4\right) + -1 \]

Reproduce

herbie shell --seed 2023047 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))