Bouland and Aaronson, Equation (26)

Average Error: 0.27% → 0.01%

Time: 7.4s

Precision: binary64

Cost: 14016

Math

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

↓

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

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) (+ (+ (pow b 4.0) (* (+ 4.0 (* 2.0 (* a a))) (* 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(a, 4.0) + ((pow(b, 4.0) + ((4.0 + (2.0 * (a * a))) * (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 = (a ** 4.0d0) + (((b ** 4.0d0) + ((4.0d0 + (2.0d0 * (a * a))) * (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(a, 4.0) + ((Math.pow(b, 4.0) + ((4.0 + (2.0 * (a * a))) * (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(a, 4.0) + ((math.pow(b, 4.0) + ((4.0 + (2.0 * (a * a))) * (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((a ^ 4.0) + Float64(Float64((b ^ 4.0) + Float64(Float64(4.0 + Float64(2.0 * Float64(a * a))) * 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 = (a ^ 4.0) + (((b ^ 4.0) + ((4.0 + (2.0 * (a * a))) * (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[Power[a, 4.0], $MachinePrecision] + N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(4.0 + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.27

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

2. Simplified 0.02

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

   [Start] 0.27	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   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. Applied egg-rr 0.28

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

4. Simplified 0.28

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

   [Start] 0.28	\[ \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right)\right) + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
   +-commutative [<=] 0.28	\[ \color{blue}{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
   distribute-lft-out [=>] 0.28	\[ \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a + b \cdot b\right)} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

5. Taylor expanded in a around 0 0.28

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

6. Simplified 0.28

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

   [Start] 0.28	\[ \left({b}^{2} + {a}^{2}\right) \cdot \left(a \cdot a + b \cdot b\right) + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
   unpow2 [=>] 0.28	\[ \left(\color{blue}{b \cdot b} + {a}^{2}\right) \cdot \left(a \cdot a + b \cdot b\right) + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
   fma-def [=>] 0.28	\[ \color{blue}{\mathsf{fma}\left(b, b, {a}^{2}\right)} \cdot \left(a \cdot a + b \cdot b\right) + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]
   unpow2 [=>] 0.28	\[ \mathsf{fma}\left(b, b, \color{blue}{a \cdot a}\right) \cdot \left(a \cdot a + b \cdot b\right) + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

7. Taylor expanded in b around 0 0.01

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

8. Simplified 0.01

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

   [Start] 0.01	\[ \left({b}^{4} + \left({a}^{4} + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)\right) - 1 \]
   associate-+r+ [=>] 0.01	\[ \color{blue}{\left(\left({b}^{4} + {a}^{4}\right) + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)} - 1 \]
   +-commutative [=>] 0.01	\[ \left(\color{blue}{\left({a}^{4} + {b}^{4}\right)} + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) - 1 \]
   associate-+r+ [<=] 0.01	\[ \color{blue}{\left({a}^{4} + \left({b}^{4} + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right)\right)} - 1 \]
   associate--l+ [=>] 0.01	\[ \color{blue}{{a}^{4} + \left(\left({b}^{4} + \left(4 + 2 \cdot {a}^{2}\right) \cdot {b}^{2}\right) - 1\right)} \]
   unpow2 [=>] 0.01	\[ {a}^{4} + \left(\left({b}^{4} + \left(4 + 2 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {b}^{2}\right) - 1\right) \]
   unpow2 [=>] 0.01	\[ {a}^{4} + \left(\left({b}^{4} + \left(4 + 2 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1\right) \]

9. Final simplification 0.01

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

Alternatives

Alternative 1
Error	0.13%
Cost	13952

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

Alternative 2
Error	0.27%
Cost	7424

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

Alternative 3
Error	2.25%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-14}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	2.37%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-14}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \end{array} \]

Alternative 5
Error	2.76%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 0.5:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6
Error	18.5%
Cost	704

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

Alternative 7
Error	35.04%
Cost	448

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

Reproduce

herbie shell --seed 2023125 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))