Average Error: 0.5 → 0.3
Time: 12.1s
Precision: 64
\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
↓
\[\mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]
x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)
↓
\mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)
double f(double x1, double x2) {
        double r41764 = x1;
        double r41765 = 2.0;
        double r41766 = r41765 * r41764;
        double r41767 = 3.0;
        double r41768 = r41767 * r41764;
        double r41769 = r41768 * r41764;
        double r41770 = x2;
        double r41771 = r41765 * r41770;
        double r41772 = r41769 + r41771;
        double r41773 = r41772 - r41764;
        double r41774 = r41764 * r41764;
        double r41775 = 1.0;
        double r41776 = r41774 + r41775;
        double r41777 = r41773 / r41776;
        double r41778 = r41766 * r41777;
        double r41779 = r41777 - r41767;
        double r41780 = r41778 * r41779;
        double r41781 = 4.0;
        double r41782 = r41781 * r41777;
        double r41783 = 6.0;
        double r41784 = r41782 - r41783;
        double r41785 = r41774 * r41784;
        double r41786 = r41780 + r41785;
        double r41787 = r41786 * r41776;
        double r41788 = r41769 * r41777;
        double r41789 = r41787 + r41788;
        double r41790 = r41774 * r41764;
        double r41791 = r41789 + r41790;
        double r41792 = r41791 + r41764;
        double r41793 = r41769 - r41771;
        double r41794 = r41793 - r41764;
        double r41795 = r41794 / r41776;
        double r41796 = r41767 * r41795;
        double r41797 = r41792 + r41796;
        double r41798 = r41764 + r41797;
        return r41798;
}
↓
double f(double x1, double x2) {
        double r41799 = 3.0;
        double r41800 = x1;
        double r41801 = 1.0;
        double r41802 = fma(r41800, r41800, r41801);
        double r41803 = r41799 / r41802;
        double r41804 = r41800 * r41800;
        double r41805 = x2;
        double r41806 = 2.0;
        double r41807 = fma(r41805, r41806, r41800);
        double r41808 = -r41807;
        double r41809 = fma(r41799, r41804, r41808);
        double r41810 = r41806 * r41805;
        double r41811 = r41810 - r41800;
        double r41812 = fma(r41799, r41804, r41811);
        double r41813 = cbrt(r41812);
        double r41814 = r41813 * r41813;
        double r41815 = r41799 * r41800;
        double r41816 = r41815 * r41800;
        double r41817 = r41813 * r41816;
        double r41818 = r41814 * r41817;
        double r41819 = r41818 / r41802;
        double r41820 = r41800 + r41819;
        double r41821 = r41816 + r41810;
        double r41822 = r41821 - r41800;
        double r41823 = r41804 + r41801;
        double r41824 = r41822 / r41823;
        double r41825 = r41824 - r41799;
        double r41826 = r41825 * r41806;
        double r41827 = r41800 * r41812;
        double r41828 = r41827 / r41802;
        double r41829 = 4.0;
        double r41830 = r41812 * r41829;
        double r41831 = r41830 / r41802;
        double r41832 = 6.0;
        double r41833 = -r41832;
        double r41834 = r41800 * r41833;
        double r41835 = fma(r41800, r41831, r41834);
        double r41836 = r41800 * r41835;
        double r41837 = fma(r41826, r41828, r41836);
        double r41838 = fma(r41800, r41804, r41800);
        double r41839 = fma(r41837, r41802, r41838);
        double r41840 = r41820 + r41839;
        double r41841 = fma(r41803, r41809, r41840);
        return r41841;
}

Error

Bits error versus x1

Bits error versus x2

Derivation

  1. Initial program 0.5

    \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-*l*0.3

    \[\leadsto \mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \color{blue}{x1 \cdot \left(x1 \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right)}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]

  5. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \color{blue}{\mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.3

    \[\leadsto \mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]

  8. Applied associate-*l*0.3

    \[\leadsto \mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)}}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]

  9. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(3, x1 \cdot x1, -\mathsf{fma}\left(x2, 2, x1\right)\right), \left(x1 + \frac{\left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) \cdot 2, \frac{x1 \cdot \mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1 \cdot x1, 2 \cdot x2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)}, x1 \cdot \left(-6\right)\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  :precision binary64
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2 x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) (- (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)) 3)) (* (* x1 x1) (- (* 4 (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) 6))) (+ (* x1 x1) 1)) (* (* (* 3 x1) x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)))) (* (* x1 x1) x1)) x1) (* 3 (/ (- (- (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))))))