Bouland and Aaronson, Equation (25)

Average Accuracy: 99.7% → 99.9%

Time: 11.0s

Precision: binary64

Cost: 21056

Math

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

↓

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

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
 (+
  (+
   (+ (fma 2.0 (* (* b b) (* a a)) (pow b 4.0)) (pow a 4.0))
   (* 4.0 (+ (* b b) (* (* a a) (+ a 1.0)))))
  -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 ((fma(2.0, ((b * b) * (a * a)), pow(b, 4.0)) + pow(a, 4.0)) + (4.0 * ((b * b) + ((a * a) * (a + 1.0))))) + -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(Float64(fma(2.0, Float64(Float64(b * b) * Float64(a * a)), (b ^ 4.0)) + (a ^ 4.0)) + Float64(4.0 * Float64(Float64(b * b) + Float64(Float64(a * a) * Float64(a + 1.0))))) + -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[(N[(2.0 * N[(N[(b * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $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. Taylor expanded in a around 0 100.0%

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

3. Simplified 100.0%

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

   [Start] 100.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   +-commutative [<=] 100.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left({b}^{4} + {a}^{4}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   associate-+r+ [=>] 100.0	\[ \left(\color{blue}{\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {b}^{4}\right) + {a}^{4}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   fma-def [=>] 100.0	\[ \left(\left(\color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right)} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 100.0	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 100.0	\[ \left(\left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   *-commutative [=>] 100.0	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

4. Taylor expanded in a around 0 99.9%

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

5. Simplified 99.9%

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

   [Start] 99.9	\[ \left(\left(\mathsf{fma}\left(2, \left(b \cdot b\right) \cdot \left(a \cdot a\right), {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + {b}^{2}\right)\right) - 1 \]
   unpow2 [=>] 99.9	\[ \left(\left(\mathsf{fma}\left(2, \left(b \cdot b\right) \cdot \left(a \cdot a\right), {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{b \cdot b}\right)\right) - 1 \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7936

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

Alternative 2
Accuracy	99.7%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + \left(b \cdot b\right) \cdot 4\right) + -1\\ \end{array} \]

Alternative 3
Accuracy	97.2%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12 + \left(b \cdot b + 4\right)\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	95.7%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12 + \left(b \cdot b + 4\right)\right) + -1\\ \end{array} \]

Alternative 5
Accuracy	95.1%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-20}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12 + \left(b \cdot b + 4\right)\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	95.5%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -51000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 55000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot -12 + \left(b \cdot b + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7
Accuracy	81.1%
Cost	960

\[\left(b \cdot b\right) \cdot \left(a \cdot -12 + \left(b \cdot b + 4\right)\right) + -1 \]

Alternative 8
Accuracy	79.6%
Cost	576

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

Alternative 9
Accuracy	64.1%
Cost	448

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

Alternative 10
Accuracy	62.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023146 
(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))