Bouland and Aaronson, Equation (24)

Percentage Accurate: 75.1% → 99.0%

Time: 9.0s

Precision: binary64

Cost: 7424

• 60.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 12\right) + -1 \]

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) (* (* b b) 12.0)) -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) + ((b * b) * 12.0)) + -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) + ((b * b) * 12.0d0)) + (-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) + ((b * b) * 12.0)) + -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) + ((b * b) * 12.0)) + -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(Float64(b * b) * 12.0)) + -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) + ((b * b) * 12.0)) + -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[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in a around 0 99.4%

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

3. Simplified 99.4%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(b \cdot \left(3 \cdot b\right)\right)}\right) - 1 \]
   Step-by-step derivation

   [Start] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(3 \cdot {b}^{2}\right)\right) - 1 \]
   *-commutative [=>] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left({b}^{2} \cdot 3\right)}\right) - 1 \]
   unpow2 [=>] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot 3\right)\right) - 1 \]
   associate-*r* [<=] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(b \cdot \left(b \cdot 3\right)\right)}\right) - 1 \]
   *-commutative [=>] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot \color{blue}{\left(3 \cdot b\right)}\right)\right) - 1 \]

4. Taylor expanded in b around 0 99.4%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

5. Simplified 99.4%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(b \cdot b\right) \cdot 12}\right) - 1 \]
   Step-by-step derivation

   [Start] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 12 \cdot {b}^{2}\right) - 1 \]
   *-commutative [=>] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{{b}^{2} \cdot 12}\right) - 1 \]
   unpow2 [=>] 99.4	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(b \cdot b\right)} \cdot 12\right) - 1 \]

6. Final simplification 99.4%

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 12\right) + -1 \]

Alternatives

Alternative 1
Accuracy	93.8%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+40}:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 2
Accuracy	82.4%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+40}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 3
Accuracy	93.9%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 34000000000000:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4
Accuracy	82.4%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -3:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1150000000000:\\ \;\;\;\;4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5
Accuracy	67.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \leq -3 \lor \neg \left(a \leq 5.5 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	71.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;a \leq -3:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot -4\right) + -1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \end{array} \]

Alternative 7
Accuracy	60.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-308} \lor \neg \left(a \leq 5.5 \cdot 10^{+147}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(a \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 8
Accuracy	51.1%
Cost	448

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

Reproduce

herbie shell --seed 2023161 
(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))