Bouland and Aaronson, Equation (26)

Average Error: 0.28% → 0.02%

Time: 8.8s

Precision: binary64

Cost: 14016

Math

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

↓

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

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

↓

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

↓

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

↓

function code(a, b)
	return Float64(Float64((a ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(2.0 * Float64(a * a))))) + Float64((b ^ 4.0) + -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 = ((a ^ 4.0) + ((b * b) * (4.0 + (2.0 * (a * a))))) + ((b ^ 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

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

Derivation

1. Initial program 0.28

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

2. Simplified 0.03

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

   [Start] 0.28	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   associate--l+ [=>] 0.28	\[ \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]
   unpow2 [=>] 0.28	\[ \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   unpow1 [<=] 0.28	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   sqr-pow [=>] 0.28	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   associate-*r* [=>] 0.2	\[ \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]
   *-commutative [<=] 0.2	\[ \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot \left(a \cdot a + b \cdot b\right)\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Taylor expanded in a around 0 0.01

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

4. Simplified 0.02

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

   [Start] 0.01	\[ \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + \left({b}^{4} + 4 \cdot {b}^{2}\right)\right)\right) - 1 \]
   sub-neg [=>] 0.01	\[ \color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + \left({b}^{4} + 4 \cdot {b}^{2}\right)\right)\right) + \left(-1\right)} \]
   metadata-eval [=>] 0.01	\[ \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + \left({b}^{4} + 4 \cdot {b}^{2}\right)\right)\right) + \color{blue}{-1} \]
   +-commutative [=>] 0.01	\[ \color{blue}{\left(\left({a}^{4} + \left({b}^{4} + 4 \cdot {b}^{2}\right)\right) + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} + -1 \]
   associate-+r+ [=>] 0.01	\[ \left(\color{blue}{\left(\left({a}^{4} + {b}^{4}\right) + 4 \cdot {b}^{2}\right)} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + -1 \]
   associate-+l+ [=>] 0.01	\[ \color{blue}{\left(\left({a}^{4} + {b}^{4}\right) + \left(4 \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} + -1 \]
   associate-*r* [=>] 0.01	\[ \left(\left({a}^{4} + {b}^{4}\right) + \left(4 \cdot {b}^{2} + \color{blue}{\left(2 \cdot {a}^{2}\right) \cdot {b}^{2}}\right)\right) + -1 \]
   distribute-rgt-in [<=] 0.01	\[ \left(\left({a}^{4} + {b}^{4}\right) + \color{blue}{{b}^{2} \cdot \left(4 + 2 \cdot {a}^{2}\right)}\right) + -1 \]
   *-commutative [<=] 0.01	\[ \left(\left({a}^{4} + {b}^{4}\right) + \color{blue}{\left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}}\right) + -1 \]

5. Applied egg-rr 0.02

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

6. Applied egg-rr 0.02

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

7. Final simplification 0.02

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

Alternatives

Alternative 1
Error	0.02%
Cost	13504

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

Alternative 2
Error	2.34%
Cost	7688

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;b \leq -0.00043:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 175:\\ \;\;\;\;\left({a}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + t_0\\ \end{array} \]

Alternative 3
Error	2.66%
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 175:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 4
Error	0.28%
Cost	7424

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

Alternative 5
Error	2.72%
Cost	7172

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left({b}^{4} + t_0\right)\\ \mathbf{elif}\;b \leq 175:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(b \cdot b\right) \cdot \left(b \cdot b + 2 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 6
Error	2.78%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \mathbf{elif}\;b \leq 175:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b + 2 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 7
Error	18.54%
Cost	960

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

Alternative 8
Error	19.46%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-1 + b \cdot \left(4 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 9
Error	19.86%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-1 + b \cdot \left(4 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 10
Error	18.54%
Cost	704

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

Alternative 11
Error	82.12%
Cost	448

\[\left(b \cdot b\right) \cdot \left(b \cdot b\right) \]

Alternative 12
Error	96.55%
Cost	320

\[4 \cdot \left(b \cdot b\right) \]

Reproduce

herbie shell --seed 2023102 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))