Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 13.3s

Precision: binary64

Cost: 39488

Math

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

↓

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

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
 (+
  (pow (hypot a b) 4.0)
  (fma 4.0 (fma a (fma a a a) (* b (* b (fma a -3.0 1.0)))) -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 pow(hypot(a, b), 4.0) + fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 1.0)))), -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((hypot(a, b) ^ 4.0) + fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), -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[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.0

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), -1\right)} \]
   Proof
   (+.f64 (pow.f64 (hypot.f64 a b) 4) (fma.f64 4 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 23 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 0 points increase in error, 23 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 19 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 0 points increase in error, 19 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 19 points increase in error, 0 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 0 points increase in error, 19 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 0 points increase in error, 0 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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (fma.f64 a a a) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a a) a)) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (+.f64 (*.f64 a a) (Rewrite<= *-lft-identity_binary64 (*.f64 1 a))) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (Rewrite<= distribute-rgt-in_binary64 (*.f64 a (+.f64 a 1))) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -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 (fma.f64 a (*.f64 a (Rewrite<= +-commutative_binary64 (+.f64 1 a))) (*.f64 b (*.f64 b (fma.f64 a -3 1)))) -1)): 19 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (fma.f64 a (*.f64 a (+.f64 1 a)) (*.f64 b (*.f64 b (fma.f64 a (Rewrite<= metadata-eval (neg.f64 3)) 1)))) -1)): 0 points increase in error, 19 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (fma.f64 a (*.f64 a (+.f64 1 a)) (*.f64 b (*.f64 b (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (neg.f64 3)) 1))))) -1)): 19 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (fma.f64 a (*.f64 a (+.f64 1 a)) (*.f64 b (*.f64 b (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 3) a)) 1)))) -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 (fma.f64 a (*.f64 a (+.f64 1 a)) (*.f64 b (*.f64 b (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 3) a)))))) -1)): 0 points increase in error, 19 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (fma.f64 a (*.f64 a (+.f64 1 a)) (*.f64 b (*.f64 b (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 3 a)))))) -1)): 19 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (fma.f64 a (*.f64 a (+.f64 1 a)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))) -1)): 0 points increase in error, 19 points decrease in error
   (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (fma.f64 4 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (*.f64 a (+.f64 1 a))) (*.f64 (*.f64 b b) (-.f64 1 (*.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 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a a) (+.f64 1 a))) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))) -1)): 4 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 1 (*.f64 3 a)))) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 4 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 1 (*.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 1 (*.f64 3 a)))))) 1)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	20864

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

Alternative 2
Error	0.2
Cost	8320

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

Alternative 3
Error	0.2
Cost	7936

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

Alternative 4
Error	1.5
Cost	7680

\[-1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + b \cdot \left(b \cdot \left(4 + a \cdot -12\right)\right)\right) \]

Alternative 5
Error	1.8
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -275:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	1.7
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -275:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 10^{-6}:\\ \;\;\;\;-1 + \left({a}^{4} + \left(a + 1\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right)\\ \end{array} \]

Alternative 7
Error	2.6
Cost	7428

\[\begin{array}{l} \mathbf{if}\;b \leq -275:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	1.5
Cost	7424

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

Alternative 9
Error	2.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -275 \lor \neg \left(b \leq 2.2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \end{array} \]

Alternative 10
Error	2.7
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -275:\\ \;\;\;\;-1 + \left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	3.1
Cost	6921

\[\begin{array}{l} \mathbf{if}\;b \leq -275 \lor \neg \left(b \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 12
Error	2.7
Cost	6793

\[\begin{array}{l} \mathbf{if}\;a \leq -95000000 \lor \neg \left(a \leq 46\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	11.9
Cost	1216

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

Alternative 14
Error	11.9
Cost	960

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

Alternative 15
Error	22.9
Cost	704

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

Alternative 16
Error	11.9
Cost	704

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

Alternative 17
Error	23.7
Cost	576

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

Alternative 18
Error	23.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022343 
(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))