Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 15.1s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (b * b)))))) - 1.0));
}

↓

double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) sqrt(((double) (((double) (a * a)) + ((double) (b * b)))))), ((double) (2.0 + 2.0)))) + ((double) (4.0 * ((double) (b * b)))))) - 1.0));
}

Error

Bits error versus a

Bits error versus b

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 0.2

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

4. Applied unpow-prod-down 0.2

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

6. Applied pow-prod-up 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020182 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))