Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 8.5s

Precision: binary64

Cost: 26816

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

↓

(FPCore (a b)
 :precision binary64
 (+
  (pow (hypot a b) 4.0)
  (fma 4.0 (fma a (- a (* a 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 pow(hypot(a, b), 4.0) + fma(4.0, fma(a, (a - (a * 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((hypot(a, b) ^ 4.0) + fma(4.0, fma(a, Float64(a - Float64(a * a)), Float64(b * Float64(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[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	21248

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

Alternative 2
Error	0.1
Cost	14976

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

Alternative 3
Error	0.2
Cost	8196

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-33}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(a \cdot a + \left(a + 3\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	8192

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

Alternative 5
Error	2.0
Cost	7812

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-33}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot \left(a \cdot a + \left(a + 3\right) \cdot \left(b \cdot b\right)\right) + {b}^{4}\right)\\ \end{array} \]

Alternative 6
Error	2.1
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-33}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]

Alternative 7
Error	2.2
Cost	7236

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-33}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + -4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]

Alternative 8
Error	2.8
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -2281193.2529852814:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 3.065916652906679 \cdot 10^{-6}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(4 + a \cdot \left(a + -4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9
Error	11.7
Cost	1600

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

Alternative 10
Error	11.7
Cost	1472

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

Alternative 11
Error	11.7
Cost	968

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + -4\right)\right)\\ \mathbf{if}\;a \leq -7313.643077613583:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.022443048435456244:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	12.4
Cost	840

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(4 + a \cdot a\right)\\ \mathbf{if}\;a \leq -7313.643077613583:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.022443048435456244:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	11.8
Cost	832

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

Alternative 14
Error	22.6
Cost	448

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

Alternative 15
Error	22.5
Cost	448

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

Alternative 16
Error	23.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022216 
(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))