Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 10.1s

Precision: binary64

Cost: 20928

Math

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

↓

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

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 4.0 (* (+ a 1.0) (* a a)) (pow a 4.0))
   (+ (* (* b b) (+ 4.0 (* (* a a) 2.0))) (pow 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(4.0, ((a + 1.0) * (a * a)), pow(a, 4.0)) + (((b * b) * (4.0 + ((a * a) * 2.0))) + pow(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(fma(4.0, Float64(Float64(a + 1.0) * Float64(a * a)), (a ^ 4.0)) + Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(a * a) * 2.0))) + (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[(4.0 * N[(N[(a + 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $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 a around 0 0.3

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

3. Simplified 0.3

   \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \color{blue}{b \cdot b}\right)\right) - 1 \]
   Proof
   (*.f64 b b): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow2_binary64 (pow.f64 b 2)): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in b around 0 0.2

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

5. Simplified 0.1

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \left(a + 1\right) \cdot \left(a \cdot a\right), {a}^{4}\right) + \left(\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + {b}^{4}\right)\right)} - 1 \]
   Proof
   (+.f64 (fma.f64 4 (*.f64 (+.f64 a 1) (*.f64 a a)) (pow.f64 a 4)) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 4 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 a)) (*.f64 a a)) (pow.f64 a 4)) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 4 (*.f64 (+.f64 1 a) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (pow.f64 a 4)) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2)) (pow.f64 a (Rewrite<= metadata-eval (*.f64 2 2)))) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (fma.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2)) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 a 2) (pow.f64 a 2)))) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 18 points increase in error, 1 points decrease in error
   (+.f64 (fma.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2)) (Rewrite<= unpow2_binary64 (pow.f64 (pow.f64 a 2) 2))) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (pow.f64 (pow.f64 a 2) 2))) (+.f64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (pow.f64 (pow.f64 a 2) 2)) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (+.f64 4 (*.f64 2 (*.f64 a a)))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (pow.f64 (pow.f64 a 2) 2)) (+.f64 (*.f64 (pow.f64 b 2) (+.f64 4 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 a 2))))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (pow.f64 (pow.f64 a 2) 2)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2))) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (pow.f64 (pow.f64 a 2) 2)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 4) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 4 (*.f64 (+.f64 1 a) (pow.f64 a 2))) (+.f64 (pow.f64 (pow.f64 a 2) 2) (+.f64 (pow.f64 b 4) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2)))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	14016

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

Alternative 2
Error	0.3
Cost	7936

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

Alternative 3
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 4
Error	2.0
Cost	7560

\[\begin{array}{l} t_0 := \left({a}^{4} + 4 \cdot \left(\left(a + 1\right) \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6500000:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.5
Cost	7424

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

Alternative 6
Error	2.8
Cost	7304

\[\begin{array}{l} t_0 := {a}^{4} + -1\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 9000000:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	2.8
Cost	7304

\[\begin{array}{l} t_0 := \left({a}^{4} + a \cdot \left(4 \cdot a\right)\right) + -1\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6500000:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	2.9
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} + -1\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6800000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	3.0
Cost	968

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7800000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	13.4
Cost	576

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

Alternative 11
Error	23.1
Cost	448

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

Reproduce

herbie shell --seed 2022325 
(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))