Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 10.6s

Precision: binary64

Cost: 27584

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (+
  (+
   (fma (fma 4.0 (+ 1.0 (* a -3.0)) (* 2.0 (* a a))) (* b b) (pow b 4.0))
   (+ (pow a 4.0) (* (* 4.0 (* a a)) (+ 1.0 a))))
  -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(fma(4.0, (1.0 + (a * -3.0)), (2.0 * (a * a))), (b * b), pow(b, 4.0)) + (pow(a, 4.0) + ((4.0 * (a * a)) * (1.0 + a)))) + -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(Float64(fma(fma(4.0, Float64(1.0 + Float64(a * -3.0)), Float64(2.0 * Float64(a * a))), Float64(b * b), (b ^ 4.0)) + Float64((a ^ 4.0) + Float64(Float64(4.0 * Float64(a * a)) * Float64(1.0 + a)))) + -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[(N[(N[(4.0 * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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. Taylor expanded in b around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(4 \cdot \left(1 - 3 \cdot a\right) + 2 \cdot {a}^{2}\right) \cdot {b}^{2} + \left({b}^{4} + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right)\right) - 1 \]
   associate-+r+ [=>] 0.0	\[ \color{blue}{\left(\left(\left(4 \cdot \left(1 - 3 \cdot a\right) + 2 \cdot {a}^{2}\right) \cdot {b}^{2} + {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right)} - 1 \]
   unpow2 [=>] 0.0	\[ \left(\left(\left(4 \cdot \left(1 - 3 \cdot a\right) + 2 \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} + {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   fma-def [=>] 0.0	\[ \left(\color{blue}{\mathsf{fma}\left(4 \cdot \left(1 - 3 \cdot a\right) + 2 \cdot {a}^{2}, b \cdot b, {b}^{4}\right)} + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   cancel-sign-sub-inv [=>] 0.0	\[ \left(\mathsf{fma}\left(4 \cdot \color{blue}{\left(1 + \left(-3\right) \cdot a\right)} + 2 \cdot {a}^{2}, b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   metadata-eval [=>] 0.0	\[ \left(\mathsf{fma}\left(4 \cdot \left(1 + \color{blue}{-3} \cdot a\right) + 2 \cdot {a}^{2}, b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   *-commutative [<=] 0.0	\[ \left(\mathsf{fma}\left(4 \cdot \left(1 + \color{blue}{a \cdot -3}\right) + 2 \cdot {a}^{2}, b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   fma-def [=>] 0.0	\[ \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot {a}^{2}\right)}, b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \color{blue}{\left(a \cdot a\right)}\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)\right) - 1 \]
   associate-*r* [=>] 0.0	\[ \left(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \left(a \cdot a\right)\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + \color{blue}{\left(4 \cdot {a}^{2}\right) \cdot \left(1 + a\right)}\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \left(a \cdot a\right)\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + \left(4 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(1 + a\right)\right)\right) - 1 \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	21440

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

Alternative 2
Error	0.1
Cost	20928

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

Alternative 3
Error	0.2
Cost	8320

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

Alternative 4
Error	0.2
Cost	7936

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

Alternative 5
Error	0.3
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-82}:\\ \;\;\;\;\left({a}^{4} + \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	1.7
Cost	7561

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016 \lor \neg \left(b \leq 0.145\right):\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right)\right) + -1\\ \end{array} \]

Alternative 7
Error	1.8
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 0.145:\\ \;\;\;\;\left({a}^{4} + \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 8
Error	2.6
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 0.145:\\ \;\;\;\;{a}^{3} \cdot \left(4 + a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 9
Error	2.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016 \lor \neg \left(b \leq 0.145\right):\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(4 + a\right) + -1\\ \end{array} \]

Alternative 10
Error	3.4
Cost	1481

\[\begin{array}{l} \mathbf{if}\;b \leq -0.017 \lor \neg \left(b \leq 8 \cdot 10^{-38}\right):\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot a\right) + -1\\ \end{array} \]

Alternative 11
Error	3.4
Cost	1225

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016 \lor \neg \left(b \leq 8 \cdot 10^{-38}\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + a \cdot -12\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot a\right) + -1\\ \end{array} \]

Alternative 12
Error	3.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;b \leq -0.016 \lor \neg \left(b \leq 8 \cdot 10^{-38}\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot a\right) + -1\\ \end{array} \]

Alternative 13
Error	12.6
Cost	704

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

Alternative 14
Error	12.7
Cost	576

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

Alternative 15
Error	22.5
Cost	448

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

Alternative 16
Error	23.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023034 
(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))