Average Error: 0.2 → 0.0
Time: 27.6s
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 r7584109 = a;
        double r7584110 = r7584109 * r7584109;
        double r7584111 = b;
        double r7584112 = r7584111 * r7584111;
        double r7584113 = r7584110 + r7584112;
        double r7584114 = 2.0;
        double r7584115 = pow(r7584113, r7584114);
        double r7584116 = 4.0;
        double r7584117 = 1.0;
        double r7584118 = r7584117 + r7584109;
        double r7584119 = r7584110 * r7584118;
        double r7584120 = 3.0;
        double r7584121 = r7584120 * r7584109;
        double r7584122 = r7584117 - r7584121;
        double r7584123 = r7584112 * r7584122;
        double r7584124 = r7584119 + r7584123;
        double r7584125 = r7584116 * r7584124;
        double r7584126 = r7584115 + r7584125;
        double r7584127 = r7584126 - r7584117;
        return r7584127;
}
↓
double f(double a, double b) {
        double r7584128 = b;
        double r7584129 = r7584128 * r7584128;
        double r7584130 = a;
        double r7584131 = r7584130 * r7584130;
        double r7584132 = r7584129 + r7584131;
        double r7584133 = sqrt(r7584132);
        double r7584134 = 4.0;
        double r7584135 = pow(r7584133, r7584134);
        double r7584136 = -3.0;
        double r7584137 = r7584136 * r7584129;
        double r7584138 = r7584130 + r7584131;
        double r7584139 = r7584137 + r7584138;
        double r7584140 = r7584139 * r7584130;
        double r7584141 = r7584140 + r7584129;
        double r7584142 = r7584141 * r7584134;
        double r7584143 = r7584135 + r7584142;
        double r7584144 = -1.0;
        double r7584145 = r7584143 + r7584144;
        return r7584145;
}

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 pow10.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied pow30.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. 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 2019139 
(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))