Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 13.3s

Precision: binary64

Cost: 21056

Math

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

↓

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

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 2.0 (* (* a b) (* a b)) (pow a 4.0)) (pow b 4.0))
   (* 4.0 (+ (* (* a a) (+ a 1.0)) (* b b))))
  -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(2.0, ((a * b) * (a * b)), pow(a, 4.0)) + pow(b, 4.0)) + (4.0 * (((a * a) * (a + 1.0)) + (b * b)))) + -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(Float64(fma(2.0, Float64(Float64(a * b) * Float64(a * b)), (a ^ 4.0)) + (b ^ 4.0)) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(b * b)))) + -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[(2.0 * N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $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. Applied egg-rr 2.1

   \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

3. Simplified 0.0

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

   [Start] 2.1	\[ \left(\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)} - 1\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-def [=>] 2.1	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   expm1-log1p [=>] 0.2	\[ \left(\color{blue}{{\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.2	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   pow-sqr [=>] 0.0	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{\left(2 \cdot 2\right)}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   metadata-eval [=>] 0.0	\[ \left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{\color{blue}{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

4. Taylor expanded in b around 0 0.0

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

5. Simplified 0.0

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

   [Start] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   +-commutative [=>] 0.0	\[ \left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \color{blue}{\left({a}^{4} + {b}^{4}\right)}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   associate-+r+ [=>] 0.0	\[ \left(\color{blue}{\left(\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + {a}^{4}\right) + {b}^{4}\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   fma-def [=>] 0.0	\[ \left(\left(\color{blue}{\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {a}^{4}\right)} + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}, {a}^{4}\right) + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unpow2 [=>] 0.0	\[ \left(\left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}, {a}^{4}\right) + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   unswap-sqr [=>] 0.0	\[ \left(\left(\mathsf{fma}\left(2, \color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}, {a}^{4}\right) + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

6. Taylor expanded in a around 0 0.1

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

7. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(\mathsf{fma}\left(2, \left(a \cdot b\right) \cdot \left(a \cdot b\right), {a}^{4}\right) + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + {b}^{2}\right)\right) - 1 \]
   unpow2 [=>] 0.1	\[ \left(\left(\mathsf{fma}\left(2, \left(a \cdot b\right) \cdot \left(a \cdot b\right), {a}^{4}\right) + {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{b \cdot b}\right)\right) - 1 \]

8. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	14016

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

Alternative 2
Error	0.2
Cost	7936

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

Alternative 3
Error	1.5
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-8}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	1.4
Cost	7680

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

Alternative 5
Error	1.6
Cost	7561

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0001 \lor \neg \left(b \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	1.7
Cost	7305

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-5} \lor \neg \left(b \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 7
Error	1.7
Cost	1480

\[\begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right) + -1\right) + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 8
Error	11.8
Cost	1216

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

Alternative 9
Error	11.8
Cost	960

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

Alternative 10
Error	11.8
Cost	704

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

Alternative 11
Error	23.6
Cost	576

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

Alternative 12
Error	12.7
Cost	576

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

Alternative 13
Error	23.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023057 
(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))