Average Error: 0.2 → 0.0
Time: 19.4s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4\]
\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(4 \cdot \left(\left(a \cdot a + b \cdot b\right) + -3 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right) + \left(-1 + {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\right) + \left(\sqrt[3]{a} \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(a \cdot a\right)\right)\right) \cdot 4
double f(double a, double b) {
        double r7878127 = a;
        double r7878128 = r7878127 * r7878127;
        double r7878129 = b;
        double r7878130 = r7878129 * r7878129;
        double r7878131 = r7878128 + r7878130;
        double r7878132 = 2.0;
        double r7878133 = pow(r7878131, r7878132);
        double r7878134 = 4.0;
        double r7878135 = 1.0;
        double r7878136 = r7878135 + r7878127;
        double r7878137 = r7878128 * r7878136;
        double r7878138 = 3.0;
        double r7878139 = r7878138 * r7878127;
        double r7878140 = r7878135 - r7878139;
        double r7878141 = r7878130 * r7878140;
        double r7878142 = r7878137 + r7878141;
        double r7878143 = r7878134 * r7878142;
        double r7878144 = r7878133 + r7878143;
        double r7878145 = r7878144 - r7878135;
        return r7878145;
}
↓
double f(double a, double b) {
        double r7878146 = 4.0;
        double r7878147 = a;
        double r7878148 = r7878147 * r7878147;
        double r7878149 = b;
        double r7878150 = r7878149 * r7878149;
        double r7878151 = r7878148 + r7878150;
        double r7878152 = -3.0;
        double r7878153 = r7878147 * r7878150;
        double r7878154 = r7878152 * r7878153;
        double r7878155 = r7878151 + r7878154;
        double r7878156 = r7878146 * r7878155;
        double r7878157 = -1.0;
        double r7878158 = sqrt(r7878151);
        double r7878159 = pow(r7878158, r7878146);
        double r7878160 = r7878157 + r7878159;
        double r7878161 = r7878156 + r7878160;
        double r7878162 = cbrt(r7878147);
        double r7878163 = r7878162 * r7878162;
        double r7878164 = r7878163 * r7878148;
        double r7878165 = r7878162 * r7878164;
        double r7878166 = r7878165 * r7878146;
        double r7878167 = r7878161 + r7878166;
        return r7878167;
}

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

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  14. Applied add-cube-cbrt0.0

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

  15. Applied associate-*r*0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(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))