Average Error: 0.2 → 0.0
Time: 9.6s
Precision: binary64
Cost: 45824
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, -3, 1\right), {a}^{3}\right)\right), -1\right) \]
(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 a (fma b (* b (fma a -3.0 1.0)) (pow a 3.0))) -1.0)))
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, a, fma(b, (b * fma(a, -3.0, 1.0)), pow(a, 3.0))), -1.0);
}

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, a, fma(b, Float64(b * fma(a, -3.0, 1.0)), (a ^ 3.0))), -1.0))
end

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

\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, a, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, -3, 1\right), {a}^{3}\right)\right), -1\right)

Error

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. Simplified0.0

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a, a, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, -3, 1\right), {a}^{3}\right)\right), -1\right)} \]
    Proof
    (+.f64 (pow.f64 (hypot.f64 a b) 4) (fma.f64 4 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (pow.f64 a 3))) -1)): 16 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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (pow.f64 a 3))) -1)): 13 points increase in error, 6 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 a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (pow.f64 a 3))) -1)): 22 points increase in error, 4 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 a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a -3 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (fma.f64 a (Rewrite<= metadata-eval (neg.f64 3)) 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (neg.f64 3)) 1))) (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 (fma.f64 a a (fma.f64 b (*.f64 b (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 3))) 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a))) 1)) (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 (fma.f64 a a (fma.f64 b (*.f64 b (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 3 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 (fma.f64 a a (fma.f64 b (*.f64 b (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 3 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 (fma.f64 a a (fma.f64 b (*.f64 b (-.f64 1 (*.f64 3 a))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 a 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 (fma.f64 a a (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b (*.f64 b (-.f64 1 (*.f64 3 a)))) (*.f64 (*.f64 a 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 (fma.f64 a a (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))) (*.f64 (*.f64 a 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 (fma.f64 a a (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 a a) 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 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a a) (+.f64 (*.f64 (*.f64 a a) 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 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 a a) (*.f64 (*.f64 a a) 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 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 a a) 1)) (*.f64 (*.f64 a a) 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=> distribute-lft-out_binary64 (*.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
    (+.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, 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 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)): 1 points increase in error, 5 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.2
Cost8320
\[-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) \]
Alternative 2
Error1.7
Cost8200
\[\begin{array}{l} \mathbf{if}\;b \leq -2.4345132204748827 \cdot 10^{-7}:\\ \;\;\;\;-1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 0.0002793985891832131:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) + {b}^{4}\right)\\ \end{array} \]
Alternative 3
Error1.7
Cost7304
\[\begin{array}{l} t_0 := -1 + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;b \leq -2.4345132204748827 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002793985891832131:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error1.7
Cost7240
\[\begin{array}{l} t_0 := -1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \mathbf{if}\;b \leq -2.4345132204748827 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002793985891832131:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error1.8
Cost1224
\[\begin{array}{l} t_0 := -1 + \left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;b \leq -2.4345132204748827 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.0002793985891832131:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error12.1
Cost832
\[-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) \]
Alternative 7
Error22.6
Cost448
\[-1 + b \cdot \left(b \cdot 4\right) \]
Alternative 8
Error23.5
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022291 
(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))