Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 11.6s

Precision: binary64

Cost: 14400

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

↓

(FPCore (a b)
 :precision binary64
 (+
  (+
   (pow (hypot b a) 4.0)
   (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (+ 1.0 (* 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) * (1.0 - (3.0 * a)))))) - 1.0;
}

↓

double code(double a, double b) {
	return (pow(hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (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) * (1.0 - (3.0 * a)))))) - 1.0;
}

↓

public static double code(double a, double b) {
	return (Math.pow(Math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (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) * (1.0 - (3.0 * a)))))) - 1.0

↓

def code(a, b):
	return (math.pow(math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (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(1.0 - Float64(3.0 * a)))))) - 1.0)
end

↓

function code(a, b)
	return Float64(Float64((hypot(b, a) ^ 4.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 + 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) * (1.0 - (3.0 * a)))))) - 1.0;
end

↓

function tmp = code(a, b)
	tmp = ((hypot(b, a) ^ 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := N[(N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 2.1

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

3. Simplified 0.0

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

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	8320

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

Alternative 2
Error	0.2
Cost	7940

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\\ \end{array} \]

Alternative 3
Error	0.2
Cost	7936

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

Alternative 4
Error	1.5
Cost	7689

\[\begin{array}{l} \mathbf{if}\;b \leq -0.086 \lor \neg \left(b \leq 1.86 \cdot 10^{-6}\right):\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \end{array} \]

Alternative 5
Error	0.2
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	1.6
Cost	7561

\[\begin{array}{l} \mathbf{if}\;b \leq -0.086 \lor \neg \left(b \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\left(b \cdot \left(b \cdot 4\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \end{array} \]

Alternative 7
Error	1.5
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -0.086:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot 4\right) + {b}^{4}\right) + -1\\ \end{array} \]

Alternative 8
Error	2.8
Cost	7305

\[\begin{array}{l} \mathbf{if}\;b \leq -0.086 \lor \neg \left(b \leq 1.9 \cdot 10^{-22}\right):\\ \;\;\;\;\left(b \cdot \left(b \cdot 4\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \end{array} \]

Alternative 9
Error	3.2
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 22000000\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \end{array} \]

Alternative 10
Error	3.6
Cost	6921

\[\begin{array}{l} \mathbf{if}\;b \leq -0.68 \lor \neg \left(b \leq 22000000\right):\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 11
Error	12.8
Cost	6656

\[{a}^{4} + -1 \]

Alternative 12
Error	22.6
Cost	704

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

Alternative 13
Error	22.6
Cost	448

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

Alternative 14
Error	23.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023041 
(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))