Average Error: 0.2 → 0.0
Time: 28.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1\]
\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(\sqrt{b \cdot b + a \cdot a}\right)}^{4} + \left(\left(-3 \cdot \left(b \cdot b\right) + \left(a + a \cdot a\right)\right) \cdot a + b \cdot b\right) \cdot 4\right) + -1
double f(double a, double b) {
        double r7192127 = a;
        double r7192128 = r7192127 * r7192127;
        double r7192129 = b;
        double r7192130 = r7192129 * r7192129;
        double r7192131 = r7192128 + r7192130;
        double r7192132 = 2.0;
        double r7192133 = pow(r7192131, r7192132);
        double r7192134 = 4.0;
        double r7192135 = 1.0;
        double r7192136 = r7192135 + r7192127;
        double r7192137 = r7192128 * r7192136;
        double r7192138 = 3.0;
        double r7192139 = r7192138 * r7192127;
        double r7192140 = r7192135 - r7192139;
        double r7192141 = r7192130 * r7192140;
        double r7192142 = r7192137 + r7192141;
        double r7192143 = r7192134 * r7192142;
        double r7192144 = r7192133 + r7192143;
        double r7192145 = r7192144 - r7192135;
        return r7192145;
}
↓
double f(double a, double b) {
        double r7192146 = b;
        double r7192147 = r7192146 * r7192146;
        double r7192148 = a;
        double r7192149 = r7192148 * r7192148;
        double r7192150 = r7192147 + r7192149;
        double r7192151 = sqrt(r7192150);
        double r7192152 = 4.0;
        double r7192153 = pow(r7192151, r7192152);
        double r7192154 = -3.0;
        double r7192155 = r7192154 * r7192147;
        double r7192156 = r7192148 + r7192149;
        double r7192157 = r7192155 + r7192156;
        double r7192158 = r7192157 * r7192148;
        double r7192159 = r7192158 + r7192147;
        double r7192160 = r7192159 * r7192152;
        double r7192161 = r7192153 + r7192160;
        double r7192162 = -1.0;
        double r7192163 = r7192161 + r7192162;
        return r7192163;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

    \[\leadsto -1 + \left(\left(b \cdot b + a \cdot \left(\left(b \cdot b\right) \cdot -3 + \left(a \cdot a + a\right)\right)\right) \cdot 4 + \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  9. Applied pow-plus0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 
(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))