Average Error: 0.2 → 0.0
Time: 20.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left(\left(b \cdot b\right) \cdot \left(3 + a\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
\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
↓
\left(\left(\left(b \cdot b\right) \cdot \left(3 + a\right) + a \cdot a\right) - a \cdot \left(a \cdot a\right)\right) \cdot 4 - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)
double f(double a, double b) {
        double r9291065 = a;
        double r9291066 = r9291065 * r9291065;
        double r9291067 = b;
        double r9291068 = r9291067 * r9291067;
        double r9291069 = r9291066 + r9291068;
        double r9291070 = 2.0;
        double r9291071 = pow(r9291069, r9291070);
        double r9291072 = 4.0;
        double r9291073 = 1.0;
        double r9291074 = r9291073 - r9291065;
        double r9291075 = r9291066 * r9291074;
        double r9291076 = 3.0;
        double r9291077 = r9291076 + r9291065;
        double r9291078 = r9291068 * r9291077;
        double r9291079 = r9291075 + r9291078;
        double r9291080 = r9291072 * r9291079;
        double r9291081 = r9291071 + r9291080;
        double r9291082 = r9291081 - r9291073;
        return r9291082;
}
↓
double f(double a, double b) {
        double r9291083 = b;
        double r9291084 = r9291083 * r9291083;
        double r9291085 = 3.0;
        double r9291086 = a;
        double r9291087 = r9291085 + r9291086;
        double r9291088 = r9291084 * r9291087;
        double r9291089 = r9291086 * r9291086;
        double r9291090 = r9291088 + r9291089;
        double r9291091 = r9291086 * r9291089;
        double r9291092 = r9291090 - r9291091;
        double r9291093 = 4.0;
        double r9291094 = r9291092 * r9291093;
        double r9291095 = 1.0;
        double r9291096 = r9291089 + r9291084;
        double r9291097 = sqrt(r9291096);
        double r9291098 = pow(r9291097, r9291093);
        double r9291099 = r9291095 - r9291098;
        double r9291100 = r9291094 - r9291099;
        return r9291100;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))