Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 10.0s

Precision: binary64

Cost: 14016

Math

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

↓

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

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) (+ (* (+ 4.0 (* 2.0 (* a a))) (* b b)) (pow a 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(b, 4.0) + (((4.0 + (2.0 * (a * a))) * (b * b)) + 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 * (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) + (((4.0d0 + (2.0d0 * (a * a))) * (b * b)) + (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 * (b * b))) - 1.0;
}

↓

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

↓

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

↓

code[a_, b_] := N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(N[(4.0 + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $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. Simplified 0.2

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

3. Taylor expanded in b around 0 0.0

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

4. Applied egg-rr 0.0

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

5. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	13952

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

Alternative 2
Error	0.2
Cost	13696

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

Alternative 3
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 4
Error	1.6
Cost	7300

\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} - 1\\ \end{array} \]

Alternative 5
Error	3.4
Cost	6920

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(\left(2 \cdot \left(a \cdot a\right) + 4\right) + b \cdot b\right) - 1\\ \mathbf{if}\;b \leq -43748.51101949656:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.250945805445222 \cdot 10^{-90}:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	2.1
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -3.6978752065879785:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.00029232286294341705:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7
Error	11.6
Cost	1088

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

Alternative 8
Error	11.7
Cost	960

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

Alternative 9
Error	12.7
Cost	576

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

Alternative 10
Error	22.4
Cost	448

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

Alternative 11
Error	23.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022320 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))