Average Error: 0.2 → 0.0
Time: 7.4s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left(2 \cdot \left(a \cdot \left(\sqrt[3]{a \cdot \left(b \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(b \cdot b\right)}\right)\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) - 1\]
\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(2 \cdot \left(a \cdot \left(\sqrt[3]{a \cdot \left(b \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(b \cdot b\right)}\right)\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) - 1
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+
   (+
    (*
     2.0
     (*
      a
      (* (cbrt (* a (* b b))) (* (cbrt (* a (* b b))) (cbrt (* a (* b b)))))))
    (+ (pow b 4.0) (pow a 4.0)))
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0)))))
  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) * (3.0 + a))))) - 1.0;
}
↓
double code(double a, double b) {
	return (((2.0 * (a * (cbrt(a * (b * b)) * (cbrt(a * (b * b)) * cbrt(a * (b * b)))))) + (pow(b, 4.0) + pow(a, 4.0))) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))) - 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

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  5. Applied add-cube-cbrt_binary64_18180.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021032 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))