Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.1% → 98.9%

Time: 11.1s

Precision: binary64

Cost: 7744

• 62.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 b b (* a a)) (+ (* a a) (* b b))) (* (* b 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(b, b, (a * a)) * ((a * a) + (b * b))) + ((b * 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(Float64(Float64(fma(b, b, Float64(a * a)) * Float64(Float64(a * a) + Float64(b * b))) + Float64(Float64(b * 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[(N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

2. Taylor expanded in a around 0 99.1%

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

3. Simplified 99.1%

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

   [Start] 99.1	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot {b}^{2}\right) - 1 \]
   unpow2 [=>] 99.1	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

4. Applied egg-rr 93.3%

   \[\leadsto \left(\color{blue}{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   Step-by-step derivation

   [Start] 99.1	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   unpow2 [=>] 99.1	\[ \left(\color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   distribute-lft-in [=>] 93.3	\[ \left(\color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-sqr-sqrt [=>] 93.3	\[ \left(\left(\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(a \cdot a\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   pow2 [=>] 93.3	\[ \left(\left(\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}} \cdot \left(a \cdot a\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   hypot-def [=>] 93.3	\[ \left(\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2} \cdot \left(a \cdot a\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-sqr-sqrt [=>] 93.3	\[ \left(\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + \color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)} \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   pow2 [=>] 93.3	\[ \left(\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}} \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   hypot-def [=>] 93.3	\[ \left(\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2} \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

5. Simplified 99.1%

   \[\leadsto \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a + b \cdot b\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   Step-by-step derivation

   [Start] 93.3	\[ \left(\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   distribute-lft-out [=>] 99.1	\[ \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a + b \cdot b\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

6. Applied egg-rr 99.1%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   Step-by-step derivation

   [Start] 99.1	\[ \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   unpow2 [=>] 99.1	\[ \left(\color{blue}{\left(\mathsf{hypot}\left(a, b\right) \cdot \mathsf{hypot}\left(a, b\right)\right)} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   hypot-udef [=>] 99.1	\[ \left(\left(\color{blue}{\sqrt{a \cdot a + b \cdot b}} \cdot \mathsf{hypot}\left(a, b\right)\right) \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   hypot-udef [=>] 99.1	\[ \left(\left(\sqrt{a \cdot a + b \cdot b} \cdot \color{blue}{\sqrt{a \cdot a + b \cdot b}}\right) \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   add-sqr-sqrt [<=] 99.1	\[ \left(\color{blue}{\left(a \cdot a + b \cdot b\right)} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   +-commutative [=>] 99.1	\[ \left(\color{blue}{\left(b \cdot b + a \cdot a\right)} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   fma-def [=>] 99.1	\[ \left(\color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)} \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

7. Final simplification 99.1%

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

Alternatives

Alternative 1
Accuracy	96.8%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;a \leq -285 \lor \neg \left(a \leq 6.9 \cdot 10^{+31}\right):\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + a \cdot -12\right)\right) + -1\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	1476

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ t_1 := a \cdot a + b \cdot b\\ \mathbf{if}\;b \cdot b \leq 0.1:\\ \;\;\;\;\left(t_0 + \left(a \cdot a\right) \cdot t_1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(b \cdot b\right) \cdot t_1\right) + -1\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	1472

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

Alternative 4
Accuracy	68.2%
Cost	960

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

Alternative 5
Accuracy	53.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \]

Alternative 6
Accuracy	68.5%
Cost	704

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

Alternative 7
Accuracy	49.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	576

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

Alternative 9
Accuracy	50.7%
Cost	448

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

Alternative 10
Accuracy	24.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023162 
(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))