Average Error: 0.2 → 0.4
Time: 5.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\left(b \cdot b\right) \cdot \left({3}^{3} + {a}^{3}\right)}{3 \cdot 3 + \left(a \cdot a - 3 \cdot a\right)}\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \frac{\left(b \cdot b\right) \cdot \left({3}^{3} + {a}^{3}\right)}{3 \cdot 3 + \left(a \cdot a - 3 \cdot a\right)}\right)\right) - 1
double code(double a, double b) {
	return ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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) * (pow(3.0, 3.0) + pow(a, 3.0))) / ((3.0 * 3.0) + ((a * a) - (3.0 * a))))))) - 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

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

  3. Applied flip3-+0.2

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

  4. Applied associate-*r/0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020058 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))