Bouland and Aaronson, Equation (24)

Average Accuracy: 99.7% → 100.0%

Time: 12.2s

Precision: binary64

Cost: 21312

Math

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

↓

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

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
 (+
  (+
   (fma 2.0 (* (* b b) (* a a)) (+ (pow b 4.0) (pow a 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 (fma(2.0, ((b * b) * (a * a)), (pow(b, 4.0) + pow(a, 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(fma(2.0, Float64(Float64(b * b) * Float64(a * a)), Float64((b ^ 4.0) + (a ^ 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

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[(2.0 * N[(N[(b * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $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 99.7%

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

2. Taylor expanded in a around 0 100.0%

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

3. Simplified 100.0%

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

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

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	20736

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

Alternative 2
Accuracy	99.7%
Cost	8192

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

Alternative 3
Accuracy	99.6%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{a}^{4} + \left(a \cdot \left(4 \cdot \left(a \cdot \left(1 - a\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + b \cdot \left(b \cdot 12\right)\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	96.8%
Cost	7561

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00046 \lor \neg \left(a \leq 5\right):\\ \;\;\;\;{a}^{4} + \left(a \cdot \left(4 \cdot \left(a \cdot \left(1 - a\right)\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + \left({b}^{4} + -1\right)\\ \end{array} \]

Alternative 5
Accuracy	95.8%
Cost	7433

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 360\right):\\ \;\;\;\;{a}^{4} + \left(-1 + a \cdot \left(\left(a \cdot a\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + \left({b}^{4} + -1\right)\\ \end{array} \]

Alternative 6
Accuracy	95.7%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 7.6\right):\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) + \left({b}^{4} + -1\right)\\ \end{array} \]

Alternative 7
Accuracy	95.6%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 620\right):\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	95.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 14.2\right):\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 9
Accuracy	95.1%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 16:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 10
Accuracy	95.1%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;a \leq -1300000000 \lor \neg \left(a \leq 26.5\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 11
Accuracy	80.8%
Cost	960

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

Alternative 12
Accuracy	63.0%
Cost	448

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

Alternative 13
Accuracy	61.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023135 
(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))