Average Error: 0.2 → 0.2
Time: 4.6s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)} \cdot \left(\sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)} \cdot \sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)}\right)\right)\right) - 1\]
\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(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)} \cdot \left(\sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)} \cdot \sqrt[3]{b \cdot \left(b \cdot \left(1 - a \cdot 3\right)\right)}\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 (+ (* a a) (* b b)) 2.0)
   (*
    4.0
    (+
     (* (* a a) (+ a 1.0))
     (*
      (cbrt (* b (* b (- 1.0 (* a 3.0)))))
      (*
       (cbrt (* b (* b (- 1.0 (* a 3.0)))))
       (cbrt (* b (* b (- 1.0 (* a 3.0))))))))))
  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) (1.0 - ((double) (3.0 * 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) (a + 1.0)))) + ((double) (((double) cbrt(((double) (b * ((double) (b * ((double) (1.0 - ((double) (a * 3.0)))))))))) * ((double) (((double) cbrt(((double) (b * ((double) (b * ((double) (1.0 - ((double) (a * 3.0)))))))))) * ((double) cbrt(((double) (b * ((double) (b * ((double) (1.0 - ((double) (a * 3.0)))))))))))))))))))) - 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 Error: 0.2 bits

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

  3. Applied add-cube-cbrtError: 0.2 bits

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

  4. SimplifiedError: 0.2 bits

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

  5. SimplifiedError: 0.2 bits

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

  6. Final simplificationError: 0.2 bits

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

Reproduce

herbie shell --seed 2020204 
(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))