Bouland and Aaronson, Equation (25)

Average Accuracy: 99.7% → 100.0%

Time: 10.5s

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 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 99.7%

   \[\leadsto \left(\color{blue}{\left(0 + {\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   Proof

   [Start] 99.7	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   add-log-exp [=>] 64.4	\[ \left(\color{blue}{\log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   *-un-lft-identity [=>] 64.4	\[ \left(\log \color{blue}{\left(1 \cdot e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   log-prod [=>] 64.4	\[ \left(\color{blue}{\left(\log 1 + \log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   metadata-eval [=>] 64.4	\[ \left(\left(\color{blue}{0} + \log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   add-log-exp [<=] 99.7	\[ \left(\left(0 + \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   add-sqr-sqrt [=>] 99.7	\[ \left(\left(0 + {\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   pow2 [=>] 99.7	\[ \left(\left(0 + {\color{blue}{\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}\right)}}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   +-commutative [=>] 99.7	\[ \left(\left(0 + {\left({\left(\sqrt{\color{blue}{b \cdot b + a \cdot a}}\right)}^{2}\right)}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   hypot-def [=>] 99.7	\[ \left(\left(0 + {\left({\color{blue}{\left(\mathsf{hypot}\left(b, a\right)\right)}}^{2}\right)}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

3. Simplified 100.0%

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

   [Start] 99.7	\[ \left(\left(0 + {\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   +-lft-identity [=>] 99.7	\[ \left(\color{blue}{{\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 99.7	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   pow-sqr [=>] 100.0	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{\left(2 \cdot 2\right)}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   metadata-eval [=>] 100.0	\[ \left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{\color{blue}{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	14016

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

Alternative 2
Accuracy	97.6%
Cost	2377

\[\begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-6} \lor \neg \left(a \leq 0.0007\right):\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	2377

\[\begin{array}{l} t_0 := a \cdot a + b \cdot b\\ t_1 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;a \leq -0.00065 \lor \neg \left(a \leq 0.0017\right):\\ \;\;\;\;\left(t_1 + \left(a \cdot a\right) \cdot t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(b \cdot b\right) \cdot t_0\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	2368

\[\begin{array}{l} t_0 := a \cdot a + b \cdot b\\ \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) + t_0 \cdot t_0\right) + -1 \end{array} \]

Alternative 5
Accuracy	95.7%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0095:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{elif}\;a \leq 170:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	79.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.235\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Accuracy	80.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 0.42\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \end{array} \]

Alternative 8
Accuracy	79.6%
Cost	704

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

Alternative 9
Accuracy	79.6%
Cost	576

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

Alternative 10
Accuracy	63.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023152 
(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))