Average Error: 0.2 → 0.0
Time: 16.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(\left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) \cdot 2 + {b}^{4}\right) + {a}^{4}\right) - 1\right) + \left(\left(a \cdot \left(a \cdot a + a\right) + b \cdot b\right) - 3 \cdot \left(\left(b \cdot b\right) \cdot a\right)\right) \cdot 4\]
\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(\left(\left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right) \cdot 2 + {b}^{4}\right) + {a}^{4}\right) - 1\right) + \left(\left(a \cdot \left(a \cdot a + a\right) + b \cdot b\right) - 3 \cdot \left(\left(b \cdot b\right) \cdot a\right)\right) \cdot 4
double f(double a, double b) {
        double r6815696 = a;
        double r6815697 = r6815696 * r6815696;
        double r6815698 = b;
        double r6815699 = r6815698 * r6815698;
        double r6815700 = r6815697 + r6815699;
        double r6815701 = 2.0;
        double r6815702 = pow(r6815700, r6815701);
        double r6815703 = 4.0;
        double r6815704 = 1.0;
        double r6815705 = r6815704 + r6815696;
        double r6815706 = r6815697 * r6815705;
        double r6815707 = 3.0;
        double r6815708 = r6815707 * r6815696;
        double r6815709 = r6815704 - r6815708;
        double r6815710 = r6815699 * r6815709;
        double r6815711 = r6815706 + r6815710;
        double r6815712 = r6815703 * r6815711;
        double r6815713 = r6815702 + r6815712;
        double r6815714 = r6815713 - r6815704;
        return r6815714;
}
↓
double f(double a, double b) {
        double r6815715 = b;
        double r6815716 = a;
        double r6815717 = r6815715 * r6815716;
        double r6815718 = r6815717 * r6815717;
        double r6815719 = 2.0;
        double r6815720 = r6815718 * r6815719;
        double r6815721 = 4.0;
        double r6815722 = pow(r6815715, r6815721);
        double r6815723 = r6815720 + r6815722;
        double r6815724 = pow(r6815716, r6815721);
        double r6815725 = r6815723 + r6815724;
        double r6815726 = 1.0;
        double r6815727 = r6815725 - r6815726;
        double r6815728 = r6815716 * r6815716;
        double r6815729 = r6815728 + r6815716;
        double r6815730 = r6815716 * r6815729;
        double r6815731 = r6815715 * r6815715;
        double r6815732 = r6815730 + r6815731;
        double r6815733 = 3.0;
        double r6815734 = r6815731 * r6815716;
        double r6815735 = r6815733 * r6815734;
        double r6815736 = r6815732 - r6815735;
        double r6815737 = r6815736 * r6815721;
        double r6815738 = r6815727 + r6815737;
        return r6815738;
}

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

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

  2. Simplified0.2

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

  3. Taylor expanded around -inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))