Average Error: 0.2 → 0.0
Time: 6.5s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(\left(b \cdot b\right) \cdot \sqrt[3]{4}\right)\right) + -1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left(2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(\left(b \cdot b\right) \cdot \sqrt[3]{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
 (+
  (+
   (+ (* 2.0 (* (* a a) (* b b))) (+ (pow b 4.0) (pow a 4.0)))
   (* (* (cbrt 4.0) (cbrt 4.0)) (* (* b b) (cbrt 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 (((2.0 * ((a * a) * (b * b))) + (pow(b, 4.0) + pow(a, 4.0))) + ((cbrt(4.0) * cbrt(4.0)) * ((b * b) * cbrt(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-cube-cbrt_binary64_18180.0

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

  6. Applied associate-*l*_binary64_17240.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2021060 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))