Bouland and Aaronson, Equation (24)

Average Accuracy: 99.7% → 100.0%

Time: 9.4s

Precision: binary64

Cost: 33152

Math

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

↓

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

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 (* b b) (+ a 3.0) (* a a)) (pow 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((b * b), (a + 3.0), (a * a)) - pow(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, Float64(fma(Float64(b * b), Float64(a + 3.0), Float64(a * a)) - (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[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Simplified 100.0%

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

   [Start] 99.7	\[ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   associate--l+ [=>] 99.7	\[ \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	27456

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

Alternative 2
Accuracy	99.7%
Cost	27008

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

Alternative 3
Accuracy	99.7%
Cost	8192

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

Alternative 4
Accuracy	97.4%
Cost	7561

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0029 \lor \neg \left(a \leq 0.000112\right):\\ \;\;\;\;-1 + \left({a}^{4} + \left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \end{array} \]

Alternative 5
Accuracy	97.4%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right)\\ \mathbf{elif}\;b \leq 0.00165:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(4, 1 - a, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.5%
Cost	7424

\[-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + b \cdot \left(b \cdot 12\right)\right) \]

Alternative 7
Accuracy	97.2%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(4, 1 - a, a \cdot a\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	97.4%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-8} \lor \neg \left(b \leq 0.00165\right):\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot 12\right) + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	97.2%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0029 \lor \neg \left(a \leq 9.2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \left(\left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b + 12, -1\right)\\ \end{array} \]

Alternative 10
Accuracy	96.4%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(\left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	94.9%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -10:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+21}:\\ \;\;\;\;-1 + \left(\left(1 - a\right) \cdot \left(a \cdot \left(a \cdot 4\right)\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 12
Accuracy	81.5%
Cost	1216

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

Alternative 13
Accuracy	80.1%
Cost	704

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

Alternative 14
Accuracy	64.5%
Cost	448

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

Alternative 15
Accuracy	64.7%
Cost	448

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

Alternative 16
Accuracy	63.2%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023136 
(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))