Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.2

Time: 1.7min

Precision: binary64

Cost: 8320

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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 b) (- 1.0 (* 3.0 a))))))
  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) * (1.0 - (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 * b) * (1.0 - (3.0 * a)))))) - 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) * (1.0d0 - (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 * b) * (1.0d0 - (3.0d0 * a)))))) - 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) * (1.0 - (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 * b) * (1.0 - (3.0 * a)))))) - 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) * (1.0 - (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 * b) * (1.0 - (3.0 * a)))))) - 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(1.0 - 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 * b) * Float64(1.0 - Float64(3.0 * a)))))) - 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) * (1.0 - (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 * b) * (1.0 - (3.0 * a)))))) - 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[(1.0 - N[(3.0 * a), $MachinePrecision]), $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 * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

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

Alternatives

Alternative 1
Error	2.1
Cost	8200

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(1 + a\right)\\ \mathbf{if}\;b \leq -3300:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot 1\right)\right) - 1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \]

Alternative 2
Error	2.0
Cost	8200

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(1 + a\right)\\ t_1 := 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -3300:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(t_0 + \left(b \cdot b\right) \cdot 1\right)\right) - 1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\left({a}^{4} + t_1\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + t_1\right) - 1\\ \end{array} \]

Alternative 3
Error	2.1
Cost	7944

\[\begin{array}{l} t_0 := \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{if}\;b \leq -3300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot 1\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.1
Cost	7944

\[\begin{array}{l} t_0 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot 1\right)\\ \mathbf{if}\;b \leq -3300:\\ \;\;\;\;\left({b}^{4} + t_0\right) - 1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\left({a}^{4} + t_0\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \]

Alternative 5
Error	3.2
Cost	7560

\[\begin{array}{l} t_0 := \left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{if}\;b \leq -3300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\left({a}^{4} + \frac{b \cdot b + a \cdot a}{0.25}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	3.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -54:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7
Error	3.1
Cost	7240

\[\begin{array}{l} \mathbf{if}\;a \leq -46:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+14}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8
Error	12.5
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -5.8:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.1:\\ \;\;\;\;4 \cdot \left(a \cdot a + b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 9
Error	11.6
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -3300:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 1.95:\\ \;\;\;\;4 \cdot \left(a \cdot a + b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 10
Error	22.0
Cost	704

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

Alternative 11
Error	22.7
Cost	448

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

Alternative 12
Error	23.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))