Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 11.3s

Precision: binary64

Cost: 14272

Math

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

↓

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

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 (hypot a b) 4.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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(hypot(a, b), 4.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) + -1.0;
}

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(Math.hypot(a, b), 4.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (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(math.hypot(a, b), 4.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (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((hypot(a, b) ^ 4.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * 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 = ((hypot(a, b) ^ 4.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (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[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $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 2.1

   \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

3. Simplified 0.0

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

   [Start] 2.1	\[ \left(\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   expm1-def [=>] 2.1	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   expm1-log1p [=>] 0.2	\[ \left(\color{blue}{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.2	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   pow-sqr [=>] 0.0	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   metadata-eval [=>] 0.0	\[ \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

4. Final simplification 0.0

   \[\leadsto \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1 \]

Alternatives

Alternative 1
Error	0.3
Cost	8196

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) + -1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	8192

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

Alternative 3
Error	1.3
Cost	8068

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + b \cdot \left(a \cdot b\right)\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	1.2
Cost	8064

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

Alternative 5
Error	1.3
Cost	7812

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + a \cdot \left(4 \cdot \left(b \cdot b\right)\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	1.3
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 7
Error	2.1
Cost	7561

\[\begin{array}{l} \mathbf{if}\;a \leq -0.038 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 8
Error	2.1
Cost	7561

\[\begin{array}{l} \mathbf{if}\;a \leq -0.038 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 9
Error	2.2
Cost	7497

\[\begin{array}{l} \mathbf{if}\;a \leq -0.038 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 10
Error	3.4
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \leq -0.14 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 11
Error	2.2
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \leq -0.047 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 4 \cdot \left(1 - a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 12
Error	4.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \leq -0.175 \lor \neg \left(a \leq 1.1 \cdot 10^{-37}\right):\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \]

Alternative 13
Error	13.0
Cost	576

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

Alternative 14
Error	22.9
Cost	448

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

Alternative 15
Error	23.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023032 
(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))