Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.0

Time: 10.3s

Precision: binary64

Cost: 66048

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(b \cdot b, a + 3, a \cdot a - {a}^{3}\right)}\\ \mathsf{fma}\left(4 \cdot {t_0}^{2}, t_0, -1 + {\left(\mathsf{hypot}\left(b, a\right)\right)}^{4}\right) \end{array} \]

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
 (let* ((t_0 (cbrt (fma (* b b) (+ a 3.0) (- (* a a) (pow a 3.0))))))
   (fma (* 4.0 (pow t_0 2.0)) t_0 (+ -1.0 (pow (hypot b a) 4.0)))))

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) {
	double t_0 = cbrt(fma((b * b), (a + 3.0), ((a * a) - pow(a, 3.0))));
	return fma((4.0 * pow(t_0, 2.0)), t_0, (-1.0 + pow(hypot(b, a), 4.0)));
}

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)
	t_0 = cbrt(fma(Float64(b * b), Float64(a + 3.0), Float64(Float64(a * a) - (a ^ 3.0))))
	return fma(Float64(4.0 * (t_0 ^ 2.0)), t_0, Float64(-1.0 + (hypot(b, a) ^ 4.0)))
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_] := Block[{t$95$0 = N[Power[N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(-1.0 + N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $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

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b, b \cdot \left(a + 3\right), a \cdot a\right) - {a}^{3}, -1\right)} \]
   Proof
   (+.f64 (pow.f64 (hypot.f64 a b) 4) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 a a) (*.f64 b b)))) 4) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 1/2)) 4) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (Rewrite<= metadata-eval (/.f64 1 2))) 4) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (Rewrite<= metadata-eval (+.f64 3 1))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) 3) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 20 points increase in error, 1 points decrease in error
   (+.f64 (*.f64 (Rewrite<= cube-unmult_binary64 (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2))))) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 13 points increase in error, 4 points decrease in error
   (+.f64 (*.f64 (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (Rewrite<= sqr-pow_binary64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 1))) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (Rewrite=> unpow1_binary64 (+.f64 (*.f64 a a) (*.f64 b b)))) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (+.f64 (*.f64 a a) (*.f64 b b))))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2)) (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (/.f64 1 2))) (+.f64 (*.f64 a a) (*.f64 b b)))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 17 points increase in error, 10 points decrease in error
   (+.f64 (*.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 1)) (+.f64 (*.f64 a a) (*.f64 b b))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> pow-plus_binary64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (+.f64 1 1))) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) (Rewrite=> metadata-eval 2)) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (+.f64 a 3)) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (-.f64 (fma.f64 b (*.f64 b (Rewrite<= +-commutative_binary64 (+.f64 3 a))) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b (*.f64 b (+.f64 3 a))) (*.f64 a a))) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (-.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 b b) (+.f64 3 a))) (*.f64 a a)) (pow.f64 a 3)) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (-.f64 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (*.f64 a a)) (Rewrite<= cube-unmult_binary64 (*.f64 a (*.f64 a a)))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (*.f64 a a)) (*.f64 (neg.f64 a) (*.f64 a a)))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (+.f64 (*.f64 a a) (*.f64 (neg.f64 a) (*.f64 a a))))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 a a))) (*.f64 (neg.f64 a) (*.f64 a a)))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (Rewrite=> distribute-rgt-out_binary64 (*.f64 (*.f64 a a) (+.f64 1 (neg.f64 a))))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (+.f64 (*.f64 (*.f64 b b) (+.f64 3 a)) (*.f64 (*.f64 a a) (Rewrite<= sub-neg_binary64 (-.f64 1 a)))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a)))) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (-.f64 1 a)) (*.f64 (*.f64 b b) (+.f64 3 a))))) 1)): 0 points increase in error, 1 points decrease in error

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	33152

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

Alternative 2
Error	0.1
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 3
Error	0.2
Cost	8196

\[\begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(a + 3\right)\\ \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-6}:\\ \;\;\;\;-1 + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + t_0\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(a \cdot a + t_0\right)\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	8192

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

Alternative 5
Error	1.4
Cost	8072

\[\begin{array}{l} \mathbf{if}\;b \leq -6.3:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;-1 + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6
Error	3.1
Cost	7560

\[\begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;-1 + \left({b}^{4} + t_0\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\left(1 - a\right) \cdot t_0 + \left(-1 + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 7
Error	3.1
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 8
Error	3.1
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + 4 \cdot a\right)\right)\\ \end{array} \]

Alternative 9
Error	1.9
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;-1 + \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + 4 \cdot a\right)\right)\\ \end{array} \]

Alternative 10
Error	3.2
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot \left(1 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \]

Alternative 11
Error	3.5
Cost	7304

\[\begin{array}{l} t_0 := -1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{if}\;a \leq -420000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 550000000000:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	2.8
Cost	7304

\[\begin{array}{l} t_0 := -1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{if}\;a \leq -420000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 550000000000:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	3.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 21000:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + -4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 14
Error	4.4
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 21000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 15
Error	3.9
Cost	6920

\[\begin{array}{l} \mathbf{if}\;a \leq -430000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 550000000000:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 16
Error	13.2
Cost	6792

\[\begin{array}{l} \mathbf{if}\;a \leq -1.95:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.5:\\ \;\;\;\;-1 + a \cdot \left(4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 17
Error	12.2
Cost	6792

\[\begin{array}{l} \mathbf{if}\;b \leq -0.29:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 18000:\\ \;\;\;\;-1 + a \cdot \left(4 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 18
Error	22.9
Cost	448

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

Alternative 19
Error	23.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022338 
(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))