Average Error: 0.2 → 0.1
Time: 24.3s
Precision: binary64
Cost: 2240
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right) + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]
\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(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right) + \left(b \cdot b\right) \cdot 12\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 (* (* b b) (* a a))) (+ (pow b 4.0) (pow a 4.0)))
   (+ (* 4.0 (* (* a a) (- 1.0 a))) (* (* b b) 12.0)))
  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 * ((b * b) * (a * a))) + (pow(b, 4.0) + pow(a, 4.0))) + ((4.0 * ((a * a) * (1.0 - a))) + ((b * b) * 12.0))) - 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

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

  4. Taylor expanded around 0 0.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \color{blue}{\left(\left(12 \cdot {b}^{2} + 4 \cdot {a}^{2}\right) - 4 \cdot {a}^{3}\right)}\right) - 1\]

  5. Simplified0.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4 + \left(b \cdot b\right) \cdot 12\right)}\right) - 1\]
  6. Using strategy rm

  7. Applied pow1_binary64_18440.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\left(\left(a \cdot a\right) \cdot \color{blue}{{\left(1 - a\right)}^{1}}\right) \cdot 4 + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

  8. Applied pow1_binary64_18440.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\left(\left(a \cdot \color{blue}{{a}^{1}}\right) \cdot {\left(1 - a\right)}^{1}\right) \cdot 4 + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

  9. Applied pow1_binary64_18440.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\left(\left(\color{blue}{{a}^{1}} \cdot {a}^{1}\right) \cdot {\left(1 - a\right)}^{1}\right) \cdot 4 + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

  10. Applied pow-prod-down_binary64_18540.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\left(\color{blue}{{\left(a \cdot a\right)}^{1}} \cdot {\left(1 - a\right)}^{1}\right) \cdot 4 + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

  11. Applied pow-prod-down_binary64_18540.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(\color{blue}{{\left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right)}^{1}} \cdot 4 + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

  12. Final simplification0.1

    \[\leadsto \left(\left(2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) + \left({b}^{4} + {a}^{4}\right)\right) + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right)\right) + \left(b \cdot b\right) \cdot 12\right)\right) - 1\]

Reproduce

herbie shell --seed 2020322 
(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))