Bouland and Aaronson, Equation (25)

Average Error: 0.3% → 0.05%

Time: 11.8s

Precision: binary64

Cost: 45824

Math

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

↓

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

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

Derivation

1. Initial program 0.3

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

2. Simplified 0.05

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

   [Start] 0.3	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   sub-neg [=>] 0.3	\[ \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) + \left(-1\right)} \]

3. Final simplification 0.05

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

Alternatives

Alternative 1
Error	0.05%
Cost	39488

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

Alternative 2
Error	0.1%
Cost	20928

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

Alternative 3
Error	0.3%
Cost	8320

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

Alternative 4
Error	0.36%
Cost	8256

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

Alternative 5
Error	0.36%
Cost	7936

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

Alternative 6
Error	2.69%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	2.31%
Cost	7680

\[\begin{array}{l} t_0 := b \cdot b + a \cdot a\\ -1 + \left({t_0}^{2} + 4 \cdot t_0\right) \end{array} \]

Alternative 8
Error	4.23%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(4 + a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 9
Error	2.84%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 10
Error	4.22%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(4 + a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(4 \cdot b\right)\right)\\ \end{array} \]

Alternative 11
Error	4.45%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;-1 + b \cdot \left(4 \cdot b + {b}^{3}\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-17}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(4 + a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	4.32%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(4 + a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	5.04%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-17}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	19.63%
Cost	1216

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

Alternative 15
Error	19.67%
Cost	960

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

Alternative 16
Error	19.69%
Cost	704

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

Alternative 17
Error	21.12%
Cost	576

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

Alternative 18
Error	37.12%
Cost	448

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

Alternative 19
Error	38.72%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023090 
(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))