Average Error: 0.2 → 0.0
Time: 11.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} + 4 \cdot \left(\left(1 \cdot \left(a \cdot a\right) + {a}^{3}\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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
↓
\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} + 4 \cdot \left(\left(1 \cdot \left(a \cdot a\right) + {a}^{3}\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r212738 = a;
        double r212739 = r212738 * r212738;
        double r212740 = b;
        double r212741 = r212740 * r212740;
        double r212742 = r212739 + r212741;
        double r212743 = 2.0;
        double r212744 = pow(r212742, r212743);
        double r212745 = 4.0;
        double r212746 = 1.0;
        double r212747 = r212746 + r212738;
        double r212748 = r212739 * r212747;
        double r212749 = 3.0;
        double r212750 = r212749 * r212738;
        double r212751 = r212746 - r212750;
        double r212752 = r212741 * r212751;
        double r212753 = r212748 + r212752;
        double r212754 = r212745 * r212753;
        double r212755 = r212744 + r212754;
        double r212756 = r212755 - r212746;
        return r212756;
}
↓
double f(double a, double b) {
        double r212757 = a;
        double r212758 = b;
        double r212759 = hypot(r212757, r212758);
        double r212760 = 2.0;
        double r212761 = 2.0;
        double r212762 = r212760 * r212761;
        double r212763 = pow(r212759, r212762);
        double r212764 = 4.0;
        double r212765 = 1.0;
        double r212766 = r212757 * r212757;
        double r212767 = r212765 * r212766;
        double r212768 = 3.0;
        double r212769 = pow(r212757, r212768);
        double r212770 = r212767 + r212769;
        double r212771 = r212758 * r212758;
        double r212772 = 3.0;
        double r212773 = r212772 * r212757;
        double r212774 = r212765 - r212773;
        double r212775 = r212771 * r212774;
        double r212776 = r212770 + r212775;
        double r212777 = r212764 * r212776;
        double r212778 = r212763 + r212777;
        double r212779 = r212778 - r212765;
        return r212779;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied unpow-prod-down0.2

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

  5. Applied fma-def0.2

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

  7. Applied fma-udef0.2

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

  8. Simplified0.0

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

  10. Applied distribute-lft-in0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))