Average Error: 0.2 → 0.0
Time: 2.8s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + \left(b \cdot b\right) \cdot 4\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4} + \left(b \cdot b\right) \cdot 4\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
 (- (+ (pow (sqrt (+ (* a a) (* b b))) 4.0) (* (* b b) 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 (pow(sqrt((a * a) + (b * b)), 4.0) + ((b * b) * 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. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.2

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

  5. Applied pow2_binary640.2

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

  6. Applied pow-pow_binary640.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020273 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))