Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 13.8s

Precision: binary64

Cost: 39488

Math

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

↓

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

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)
  (fma 4.0 (fma a (fma a a a) (* b (* b (fma a -3.0 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 pow(hypot(a, b), 4.0) + fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 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((hypot(a, b) ^ 4.0) + fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 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[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

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

   [Start] 0.2	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   associate--l+ [=>] 0.2	\[ \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]
   unpow2 [=>] 0.2	\[ \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   unpow1 [<=] 0.2	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   sqr-pow [=>] 0.2	\[ \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   associate-*r* [=>] 0.1	\[ \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   *-commutative [=>] 0.1	\[ \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot \left(a \cdot a + b \cdot b\right)\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   unpow1 [<=] 0.1	\[ \left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   sqr-pow [=>] 0.1	\[ \left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   cube-unmult [=>] 0.1	\[ \color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{3}} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   pow-plus [=>] 0.0	\[ \color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{\left(3 + 1\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   metadata-eval [=>] 0.0	\[ {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   unpow1/2 [=>] 0.0	\[ {\color{blue}{\left(\sqrt{a \cdot a + b \cdot b}\right)}}^{\left(3 + 1\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   hypot-def [=>] 0.0	\[ {\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{\left(3 + 1\right)} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   metadata-eval [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{\color{blue}{4}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]
   fma-neg [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right), -1\right)} \]
   associate-*l* [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{a \cdot \left(a \cdot \left(1 + a\right)\right)} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right), -1\right) \]
   fma-def [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \color{blue}{\mathsf{fma}\left(a, a \cdot \left(1 + a\right), \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)}, -1\right) \]
   +-commutative [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot \color{blue}{\left(a + 1\right)}, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), -1\right) \]
   distribute-rgt-in [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \color{blue}{a \cdot a + 1 \cdot a}, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), -1\right) \]
   *-lft-identity [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a \cdot a + \color{blue}{a}, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), -1\right) \]
   fma-def [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, a, a\right)}, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), -1\right) \]
   associate-*l* [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \color{blue}{b \cdot \left(b \cdot \left(1 - 3 \cdot a\right)\right)}\right), -1\right) \]
   cancel-sign-sub-inv [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \color{blue}{\left(1 + \left(-3\right) \cdot a\right)}\right)\right), -1\right) \]
   +-commutative [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \color{blue}{\left(\left(-3\right) \cdot a + 1\right)}\right)\right), -1\right) \]
   *-commutative [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \left(\color{blue}{a \cdot \left(-3\right)} + 1\right)\right)\right), -1\right) \]
   fma-def [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(a, -3, 1\right)}\right)\right), -1\right) \]
   metadata-eval [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, \color{blue}{-3}, 1\right)\right)\right), -1\right) \]
   metadata-eval [=>] 0.0	\[ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), \color{blue}{-1}\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	14400

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

Alternative 2
Error	0.2
Cost	8704

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

Alternative 3
Error	0.2
Cost	8320

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

Alternative 4
Error	1.1
Cost	7944

\[\begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-59}:\\ \;\;\;\;-1 + \left(\left(a \cdot 4\right) \cdot \left(a + a \cdot a\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 5
Error	0.2
Cost	7936

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

Alternative 6
Error	1.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-59}:\\ \;\;\;\;-1 + \left(\left(a \cdot 4\right) \cdot \left(a + a \cdot a\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + b \cdot \left(b \cdot 4\right)\right)\\ \end{array} \]

Alternative 7
Error	1.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 8
Error	1.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + \left(\left(a \cdot 4\right) \cdot \left(a + a \cdot a\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 9
Error	1.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + \left(\left(a \cdot 4\right) \cdot \left(a + a \cdot a\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 10
Error	3.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 11
Error	3.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 12
Error	3.4
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 13
Error	3.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;\left(1 + a \cdot a\right) \cdot \left(-1 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 14
Error	14.0
Cost	704

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

Alternative 15
Error	23.2
Cost	448

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

Alternative 16
Error	23.0
Cost	448

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

Alternative 17
Error	24.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022349 
(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))