Average Error: 0.2 → 0.1
Time: 1.9min
Precision: binary64
Cost: 14336
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + 4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + 4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(a + 1\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) (- 1.0 (* 3.0 a))))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (sqrt (+ (* a a) (* b b))) 4.0)
   (* 4.0 (+ (* b b) (* (* a a) (+ a 1.0)))))
  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) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return (pow(sqrt((a * a) + (b * b)), 4.0) + (4.0 * ((b * b) + ((a * a) * (a + 1.0))))) - 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

Alternatives

Alternative 1
Error0.2
Cost1984
\[\left(4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) + \left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right) - 1\]
Alternative 2
Error1.4
Cost1728
\[\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(a \cdot a + b \cdot b\right) \cdot 4\right) - 1\]
Alternative 3
Error1.5
Cost1472
\[\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 4\right) + -1\]
Alternative 4
Error2.8
Cost1544
\[\begin{array}{l} \mathbf{if}\;a \leq -878993.3945537906 \lor \neg \left(a \leq 182.58234965794426\right):\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + \left(a \cdot a\right) \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(b \cdot b\right) \cdot 4 + \left(b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right)\\ \end{array}\]
Alternative 5
Error11.7
Cost1216
\[-1 + \left(\left(b \cdot b\right) \cdot 4 + \left(b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)\right)\]
Alternative 6
Error23.6
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.2

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

  3. Applied unpow2_binary64_21890.2

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  7. Applied pow1_binary64_21850.2

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

  8. Applied pow1_binary64_21850.2

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

  9. Applied pow-prod-down_binary64_21950.2

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

  10. Simplified0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2021014 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))