Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.2

Time: 1.8min

Precision: binary64

Cost: 8576

Math

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

↓

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{b \cdot \left(27 \cdot b\right)}{9 + a \cdot \left(a + -3\right)}\right)\right) - 1 \]

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
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (*
    4.0
    (+ (* (* a a) (- 1.0 a)) (/ (* b (* 27.0 b)) (+ 9.0 (* a (+ a -3.0)))))))
  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 (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * (27.0 * b)) / (9.0 + (a * (a + -3.0))))))) - 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 = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * (27.0d0 * b)) / (9.0d0 + (a * (a + (-3.0d0)))))))) - 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 (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * (27.0 * b)) / (9.0 + (a * (a + -3.0))))))) - 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 (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * (27.0 * b)) / (9.0 + (a * (a + -3.0))))))) - 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((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * Float64(27.0 * b)) / Float64(9.0 + Float64(a * Float64(a + -3.0))))))) - 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 = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * (27.0 * b)) / (9.0 + (a * (a + -3.0))))))) - 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[(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 * N[(27.0 * b), $MachinePrecision]), $MachinePrecision] / N[(9.0 + N[(a * N[(a + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.4

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \color{blue}{\frac{\left(b \cdot \left(27 + {a}^{3}\right)\right) \cdot b}{9 + a \cdot \left(a + -3\right)}}\right)\right) - 1 \]

3. Taylor expanded in a around 0 0.2

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\color{blue}{27 \cdot {b}^{2}}}{9 + a \cdot \left(a + -3\right)}\right)\right) - 1 \]

4. Simplified 0.2

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\color{blue}{b \cdot \left(27 \cdot b\right)}}{9 + a \cdot \left(a + -3\right)}\right)\right) - 1 \]
   Proof

Alternatives

Alternative 1
Error	1.6
Cost	8456

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(1 - a\right)\\ \mathbf{if}\;a \leq -0.0125:\\ \;\;\;\;\left({a}^{4} + 4 \cdot t_0\right) - 1\\ \mathbf{elif}\;a \leq 0.00076:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(t_0 + \frac{b \cdot \left(27 \cdot b\right)}{9 + a \cdot \left(a + -3\right)}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	8192

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

Alternative 3
Error	1.6
Cost	8072

\[\begin{array}{l} \mathbf{if}\;b \leq -3300:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 \cdot a + 12\right)\right) - 1\\ \mathbf{elif}\;b \leq 0.245:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 12 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \]

Alternative 4
Error	1.6
Cost	8072

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(1 - a\right)\\ t_1 := 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\\ \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;\left({a}^{4} + 4 \cdot t_0\right) - 1\\ \mathbf{elif}\;a \leq 0.00023:\\ \;\;\;\;\left({b}^{4} + t_1\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + t_1\right) - 1\\ \end{array} \]

Alternative 5
Error	1.8
Cost	7560

\[\begin{array}{l} t_0 := a \cdot \left({a}^{3} + 4 \cdot \left(a \cdot \left(1 - a\right)\right)\right) - 1\\ \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\left({b}^{4} + 12 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.8
Cost	7560

\[\begin{array}{l} t_0 := \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) - 1\\ \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\left({b}^{4} + 12 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.8
Cost	7560

\[\begin{array}{l} t_0 := \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) - 1\\ \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00024:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 \cdot a + 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	3.0
Cost	7304

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -2.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.0007:\\ \;\;\;\;\left({b}^{4} + 12 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	3.0
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -1.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00058:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(12 + b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	12.0
Cost	704

\[\left(b \cdot b\right) \cdot \left(12 + b \cdot b\right) - 1 \]

Alternative 11
Error	22.7
Cost	448

\[12 \cdot \left(b \cdot b\right) - 1 \]

Alternative 12
Error	23.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023033 
(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))