Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({b}^{2} \cdot 12 + \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + {a}^{2} \cdot \left(4 - a \cdot 4\right)\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({b}^{2} \cdot 12 + \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + {a}^{2} \cdot \left(4 - a \cdot 4\right)\right)\right) - 1
double f(double a, double b) {
        double r406089 = a;
        double r406090 = r406089 * r406089;
        double r406091 = b;
        double r406092 = r406091 * r406091;
        double r406093 = r406090 + r406092;
        double r406094 = 2.0;
        double r406095 = pow(r406093, r406094);
        double r406096 = 4.0;
        double r406097 = 1.0;
        double r406098 = r406097 - r406089;
        double r406099 = r406090 * r406098;
        double r406100 = 3.0;
        double r406101 = r406100 + r406089;
        double r406102 = r406092 * r406101;
        double r406103 = r406099 + r406102;
        double r406104 = r406096 * r406103;
        double r406105 = r406095 + r406104;
        double r406106 = r406105 - r406097;
        return r406106;
}
↓
double f(double a, double b) {
        double r406107 = b;
        double r406108 = 2.0;
        double r406109 = pow(r406107, r406108);
        double r406110 = 12.0;
        double r406111 = r406109 * r406110;
        double r406112 = a;
        double r406113 = r406107 * r406107;
        double r406114 = fma(r406112, r406112, r406113);
        double r406115 = 2.0;
        double r406116 = pow(r406114, r406115);
        double r406117 = pow(r406112, r406108);
        double r406118 = 4.0;
        double r406119 = r406112 * r406118;
        double r406120 = r406118 - r406119;
        double r406121 = r406117 * r406120;
        double r406122 = r406116 + r406121;
        double r406123 = r406111 + r406122;
        double r406124 = 1.0;
        double r406125 = r406123 - r406124;
        return r406125;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left(\left(4 \cdot {a}^{2} + 12 \cdot {b}^{2}\right) - 4 \cdot {a}^{3}\right)}\right) - 1\]

  3. Simplified0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\mathsf{fma}\left({a}^{2}, 4, 12 \cdot {b}^{2} - 4 \cdot {a}^{3}\right)}\right) - 1\]
  4. Using strategy rm

  5. Applied fma-udef0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{\left({a}^{2} \cdot 4 + \left(12 \cdot {b}^{2} - 4 \cdot {a}^{3}\right)\right)}\right) - 1\]

  6. Applied associate-+r+0.2

    \[\leadsto \color{blue}{\left(\left({\left(a \cdot a + b \cdot b\right)}^{2} + {a}^{2} \cdot 4\right) + \left(12 \cdot {b}^{2} - 4 \cdot {a}^{3}\right)\right)} - 1\]

  7. Simplified0.2

    \[\leadsto \left(\color{blue}{\mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right)} + \left(12 \cdot {b}^{2} - 4 \cdot {a}^{3}\right)\right) - 1\]
  8. Using strategy rm

  9. Applied sub-neg0.2

    \[\leadsto \left(\mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \color{blue}{\left(12 \cdot {b}^{2} + \left(-4 \cdot {a}^{3}\right)\right)}\right) - 1\]

  10. Applied associate-+r+0.2

    \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + 12 \cdot {b}^{2}\right) + \left(-4 \cdot {a}^{3}\right)\right)} - 1\]

  11. Simplified0.2

    \[\leadsto \left(\color{blue}{\mathsf{fma}\left({b}^{2}, 12, \mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)} + \left(-4 \cdot {a}^{3}\right)\right) - 1\]
  12. Using strategy rm

  13. Applied fma-udef0.2

    \[\leadsto \left(\color{blue}{\left({b}^{2} \cdot 12 + \mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)} + \left(-4 \cdot {a}^{3}\right)\right) - 1\]

  14. Applied associate-+l+0.2

    \[\leadsto \color{blue}{\left({b}^{2} \cdot 12 + \left(\mathsf{fma}\left(4, {a}^{2}, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(-4 \cdot {a}^{3}\right)\right)\right)} - 1\]

  15. Simplified0.2

    \[\leadsto \left({b}^{2} \cdot 12 + \color{blue}{\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + {a}^{2} \cdot \left(4 - a \cdot 4\right)\right)}\right) - 1\]

  16. Final simplification0.2

    \[\leadsto \left({b}^{2} \cdot 12 + \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + {a}^{2} \cdot \left(4 - a \cdot 4\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))