Average Error: 0.2 → 0.3
Time: 28.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \left(\sqrt{\sqrt{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}}} \cdot \sqrt{\sqrt{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}}}\right) \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}}, \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1\]
\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
↓
\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \left(\sqrt{\sqrt{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}}} \cdot \sqrt{\sqrt{\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}}}\right) \cdot \sqrt{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}}, \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(1 + a\right) \cdot \left(a \cdot a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r12854786 = a;
        double r12854787 = r12854786 * r12854786;
        double r12854788 = b;
        double r12854789 = r12854788 * r12854788;
        double r12854790 = r12854787 + r12854789;
        double r12854791 = 2.0;
        double r12854792 = pow(r12854790, r12854791);
        double r12854793 = 4.0;
        double r12854794 = 1.0;
        double r12854795 = r12854794 + r12854786;
        double r12854796 = r12854787 * r12854795;
        double r12854797 = 3.0;
        double r12854798 = r12854797 * r12854786;
        double r12854799 = r12854794 - r12854798;
        double r12854800 = r12854789 * r12854799;
        double r12854801 = r12854796 + r12854800;
        double r12854802 = r12854793 * r12854801;
        double r12854803 = r12854792 + r12854802;
        double r12854804 = r12854803 - r12854794;
        return r12854804;
}
↓
double f(double a, double b) {
        double r12854805 = a;
        double r12854806 = r12854805 * r12854805;
        double r12854807 = b;
        double r12854808 = r12854807 * r12854807;
        double r12854809 = r12854806 + r12854808;
        double r12854810 = 2.0;
        double r12854811 = pow(r12854809, r12854810);
        double r12854812 = sqrt(r12854811);
        double r12854813 = cbrt(r12854811);
        double r12854814 = r12854813 * r12854813;
        double r12854815 = sqrt(r12854814);
        double r12854816 = sqrt(r12854815);
        double r12854817 = sqrt(r12854813);
        double r12854818 = sqrt(r12854817);
        double r12854819 = r12854816 * r12854818;
        double r12854820 = sqrt(r12854812);
        double r12854821 = r12854819 * r12854820;
        double r12854822 = 1.0;
        double r12854823 = 3.0;
        double r12854824 = r12854823 * r12854805;
        double r12854825 = r12854822 - r12854824;
        double r12854826 = r12854808 * r12854825;
        double r12854827 = r12854822 + r12854805;
        double r12854828 = r12854827 * r12854806;
        double r12854829 = r12854826 + r12854828;
        double r12854830 = 4.0;
        double r12854831 = r12854829 * r12854830;
        double r12854832 = fma(r12854812, r12854821, r12854831);
        double r12854833 = r12854832 - r12854822;
        return r12854833;
}

Error

Bits error versus a

Bits error versus b

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied fma-def0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.2

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

  9. Applied add-cube-cbrt0.3

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

  10. Applied sqrt-prod0.3

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

  11. Applied sqrt-prod0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))