Average Error: 0.2 → 0.0
Time: 4.5s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left({a}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\right) + {b}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left({a}^{4} + \left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right)\right) + {b}^{4}\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
 (- (+ (+ (pow a 4.0) (* (* b b) (+ 4.0 (* 2.0 (* a a))))) (pow 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 ((pow(a, 4.0) + ((b * b) * (4.0 + (2.0 * (a * a))))) + pow(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

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

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

  5. Applied add-cbrt-cube_binary6416.4

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

  6. Simplified16.4

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

  7. Taylor expanded around 0 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2021174 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))