Average Error: 0.2 → 0.0
Time: 1.4min
Precision: binary64
Cost: 14272
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + b \cdot \left(b \cdot 4\right)\right) + -1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + b \cdot \left(b \cdot 4\right)\right) + -1
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (+
   (+ (* (* a a) (* b (* b 2.0))) (+ (pow b 4.0) (pow a 4.0)))
   (* b (* b 4.0)))
  -1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}
↓
double code(double a, double b) {
	return ((((a * a) * (b * (b * 2.0))) + (pow(b, 4.0) + pow(a, 4.0))) + (b * (b * 4.0))) + -1.0;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.1
Cost7680
\[-1 + \left({a}^{4} + \left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2 + \left(4 + b \cdot b\right)\right)\right)\]
Alternative 2
Error0.2
Cost1472
\[-1 + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + 4 \cdot \left(b \cdot b\right)\right)\]
Alternative 3
Error12.1
Cost1088
\[-1 + b \cdot \left(b \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\right)\]
Alternative 4
Error12.1
Cost1088
\[-1 + \left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\]
Alternative 5
Error12.2
Cost704
\[-1 + \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\]
Alternative 6
Error13.2
Cost704
\[\left(b \cdot b + 1\right) \cdot \left(-1 + b \cdot b\right)\]
Alternative 7
Error13.2
Cost576
\[-1 + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\]
Alternative 8
Error23.7
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2021042 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))