Bouland and Aaronson, Equation (24)

Average Error: 0.29% → 0.03%

Time: 11.1s

Precision: binary64

Cost: 21312

Math

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

↓

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

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(Float64(fma(2.0, Float64(Float64(b * b) * Float64(a * a)), (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[(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[(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 0.29

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

2. Taylor expanded in a around 0 0.03

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

3. Simplified 0.03

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

   [Start] 0.03	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   +-commutative [=>] 0.03	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left({b}^{4} + {a}^{4}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   associate-+r+ [=>] 0.03	\[ \left(\color{blue}{\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {b}^{4}\right) + {a}^{4}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   fma-def [=>] 0.03	\[ \left(\left(\color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right)} + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   *-commutative [=>] 0.03	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{{b}^{2} \cdot {a}^{2}}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.03	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.03	\[ \left(\left(\mathsf{fma}\left(2, \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}, {b}^{4}\right) + {a}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

4. Final simplification 0.03

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

Alternatives

Alternative 1
Error	0.29%
Cost	8576

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

Alternative 2
Error	0.29%
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
Error	1.07%
Cost	7944

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right) + -1\\ \mathbf{elif}\;b \cdot b \leq 10000000000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(12 + \left(b \cdot b + a \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(b \cdot b + a \cdot a\right)}^{2} + a \cdot \left(a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	2.33%
Cost	7936

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

Alternative 5
Error	2.69%
Cost	7556

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

Alternative 6
Error	4.18%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot 12\right) + -1\\ \end{array} \]

Alternative 7
Error	4.16%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;{a}^{3} \cdot \left(a + -4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(12 + \left(b \cdot b + a \cdot 4\right)\right) + \left(b \cdot b\right) \cdot \left(a \cdot \left(2 \cdot a\right)\right)\right) + -1\\ \end{array} \]

Alternative 8
Error	4.87%
Cost	6916

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

Alternative 9
Error	4.87%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;a \leq -13.5 \lor \neg \left(a \leq 1.35 \cdot 10^{+14}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(12 + \left(b \cdot b + a \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 10
Error	18.52%
Cost	1600

\[\left(\left(b \cdot b\right) \cdot \left(12 + \left(b \cdot b + a \cdot 4\right)\right) + \left(b \cdot b\right) \cdot \left(a \cdot \left(2 \cdot a\right)\right)\right) + -1 \]

Alternative 11
Error	18.72%
Cost	704

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

Alternative 12
Error	35.49%
Cost	448

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

Alternative 13
Error	37.1%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023121 
(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))