Average Error: 0.2 → 0.2
Time: 3.4s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right)} \cdot \sqrt{{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right)} - 1\]
\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
↓
\sqrt{{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right)} \cdot \sqrt{{\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(a + 3\right) + \left(a \cdot a\right) \cdot \left(1 - a\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
 (-
  (*
   (sqrt
    (+
     (pow (+ (* b b) (* a a)) 2.0)
     (* 4.0 (+ (* (* b b) (+ a 3.0)) (* (* a a) (- 1.0 a))))))
   (sqrt
    (+
     (pow (+ (* b b) (* a a)) 2.0)
     (* 4.0 (+ (* (* b b) (+ a 3.0)) (* (* a a) (- 1.0 a)))))))
  1.0))
double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (((double) (((double) (a * a)) * ((double) (1.0 - a)))) + ((double) (((double) (b * b)) * ((double) (3.0 + a)))))))))) - 1.0));
}
↓
double code(double a, double b) {
	return ((double) (((double) (((double) sqrt(((double) (((double) pow(((double) (((double) (b * b)) + ((double) (a * a)))), 2.0)) + ((double) (4.0 * ((double) (((double) (((double) (b * b)) * ((double) (a + 3.0)))) + ((double) (((double) (a * a)) * ((double) (1.0 - a)))))))))))) * ((double) sqrt(((double) (((double) pow(((double) (((double) (b * b)) + ((double) (a * a)))), 2.0)) + ((double) (4.0 * ((double) (((double) (((double) (b * b)) * ((double) (a + 3.0)))) + ((double) (((double) (a * a)) * ((double) (1.0 - a)))))))))))))) - 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. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020210 
(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))