Average Error: 0.2 → 0.2
Time: 6.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\left({\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
↓
\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2}}, \sqrt{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {\left(\mathsf{hypot}\left(a, b\right)\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
double f(double a, double b) {
        double r287942 = a;
        double r287943 = r287942 * r287942;
        double r287944 = b;
        double r287945 = r287944 * r287944;
        double r287946 = r287943 + r287945;
        double r287947 = 2.0;
        double r287948 = pow(r287946, r287947);
        double r287949 = 4.0;
        double r287950 = 1.0;
        double r287951 = r287950 - r287942;
        double r287952 = r287943 * r287951;
        double r287953 = 3.0;
        double r287954 = r287953 + r287942;
        double r287955 = r287945 * r287954;
        double r287956 = r287952 + r287955;
        double r287957 = r287949 * r287956;
        double r287958 = r287948 + r287957;
        double r287959 = r287958 - r287950;
        return r287959;
}
↓
double f(double a, double b) {
        double r287960 = a;
        double r287961 = r287960 * r287960;
        double r287962 = b;
        double r287963 = r287962 * r287962;
        double r287964 = r287961 + r287963;
        double r287965 = 2.0;
        double r287966 = pow(r287964, r287965);
        double r287967 = sqrt(r287966);
        double r287968 = hypot(r287960, r287962);
        double r287969 = pow(r287968, r287965);
        double r287970 = r287969 * r287969;
        double r287971 = sqrt(r287970);
        double r287972 = 4.0;
        double r287973 = 1.0;
        double r287974 = r287973 - r287960;
        double r287975 = r287961 * r287974;
        double r287976 = 3.0;
        double r287977 = r287976 + r287960;
        double r287978 = r287963 * r287977;
        double r287979 = r287975 + r287978;
        double r287980 = r287972 * r287979;
        double r287981 = fma(r287967, r287971, r287980);
        double r287982 = r287981 - r287973;
        return r287982;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied fma-def0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied unpow-prod-down0.2

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 +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))