Bouland and Aaronson, Equation (25)

Average Accuracy: 99.7% → 100.0%

Time: 13.6s

Precision: binary64

Cost: 14400

Math

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

↓

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

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 a b) 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(a, b), 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(a, b), 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(a, b), 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(a, b) ^ 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(a, b) ^ 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[a ^ 2 + b ^ 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 99.7%

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

2. Applied egg-rr 96.6%

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

   [Start] 99.7	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-log1p-u [=>] 96.6	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-udef [=>] 96.6	\[ \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)} - 1\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   add-sqr-sqrt [=>] 96.6	\[ \left(\left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   pow2 [=>] 96.6	\[ \left(\left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}\right)}}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   hypot-def [=>] 96.6	\[ \left(\left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

3. Simplified 100.0%

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

   [Start] 96.6	\[ \left(\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-def [=>] 96.6	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-log1p [=>] 99.7	\[ \left(\color{blue}{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 99.7	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   pow-sqr [=>] 100.0	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   metadata-eval [=>] 100.0	\[ \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	8196

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

Alternative 2
Accuracy	99.6%
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 3
Accuracy	99.6%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-38}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	7561

\[\begin{array}{l} \mathbf{if}\;a \leq -0.115 \lor \neg \left(a \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \end{array} \]

Alternative 5
Accuracy	95.7%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	94.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]

Alternative 7
Accuracy	94.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]

Alternative 8
Accuracy	80.5%
Cost	2249

\[\begin{array}{l} \mathbf{if}\;a \leq -4 \lor \neg \left(a \leq 1.9\right):\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) + -1\\ \end{array} \]

Alternative 9
Accuracy	80.5%
Cost	1865

\[\begin{array}{l} \mathbf{if}\;a \leq -4 \lor \neg \left(a \leq 2.1\right):\\ \;\;\;\;a \cdot \left(\left(a \cdot a\right) \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\right) + -1\\ \end{array} \]

Alternative 10
Accuracy	78.8%
Cost	704

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

Alternative 11
Accuracy	63.0%
Cost	448

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

Reproduce

herbie shell --seed 2023135 
(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))