Average Error: 0.2 → 0.1
Time: 6.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)}} \cdot {\left(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)}}\right)}^{3} - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)}} \cdot {\left(\sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)}}\right)}^{3} - 1
double f(double a, double b) {
        double r291811 = a;
        double r291812 = r291811 * r291811;
        double r291813 = b;
        double r291814 = r291813 * r291813;
        double r291815 = r291812 + r291814;
        double r291816 = 2.0;
        double r291817 = pow(r291815, r291816);
        double r291818 = 4.0;
        double r291819 = 1.0;
        double r291820 = r291819 - r291811;
        double r291821 = r291812 * r291820;
        double r291822 = 3.0;
        double r291823 = r291822 + r291811;
        double r291824 = r291814 * r291823;
        double r291825 = r291821 + r291824;
        double r291826 = r291818 * r291825;
        double r291827 = r291817 + r291826;
        double r291828 = r291827 - r291819;
        return r291828;
}
↓
double f(double a, double b) {
        double r291829 = a;
        double r291830 = r291829 * r291829;
        double r291831 = b;
        double r291832 = r291831 * r291831;
        double r291833 = r291830 + r291832;
        double r291834 = 2.0;
        double r291835 = pow(r291833, r291834);
        double r291836 = 4.0;
        double r291837 = 1.0;
        double r291838 = r291837 - r291829;
        double r291839 = r291830 * r291838;
        double r291840 = 3.0;
        double r291841 = r291840 + r291829;
        double r291842 = r291831 * r291841;
        double r291843 = r291831 * r291842;
        double r291844 = r291839 + r291843;
        double r291845 = r291836 * r291844;
        double r291846 = r291835 + r291845;
        double r291847 = sqrt(r291846);
        double r291848 = sqrt(r291847);
        double r291849 = 3.0;
        double r291850 = pow(r291848, r291849);
        double r291851 = r291848 * r291850;
        double r291852 = r291851 - r291837;
        return r291852;
}

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. Using strategy rm

  3. Applied associate-*l*0.2

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

  5. Applied add-sqr-sqrt0.2

    \[\leadsto \color{blue}{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + b \cdot \left(b \cdot \left(3 + a\right)\right)\right)}} - 1\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.2

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

  9. Applied associate-*l*0.2

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))