Average Error: 0.2 → 0.0
Time: 2.4s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(2 \cdot {\left(a \cdot b\right)}^{2} + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left(2 \cdot {\left(a \cdot b\right)}^{2} + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\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
 (-
  (+ (+ (* 2.0 (pow (* a b) 2.0)) (+ (pow b 4.0) (pow a 4.0))) (* 4.0 (* b b)))
  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 (((2.0 * pow((a * b), 2.0)) + (pow(b, 4.0) + pow(a, 4.0))) + (4.0 * (b * b))) - 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 pow2_binary640.0

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

  6. Applied pow2_binary640.0

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

  7. Applied pow-prod-down_binary640.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2021173 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))