Average Error: 0.2 → 0.2
Time: 18.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({a}^{2}, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left({a}^{2}, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\]
\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
↓
\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({a}^{2}, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left({a}^{2}, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1
double f(double a, double b) {
        double r150092 = a;
        double r150093 = r150092 * r150092;
        double r150094 = b;
        double r150095 = r150094 * r150094;
        double r150096 = r150093 + r150095;
        double r150097 = 2.0;
        double r150098 = pow(r150096, r150097);
        double r150099 = 4.0;
        double r150100 = 1.0;
        double r150101 = r150100 + r150092;
        double r150102 = r150093 * r150101;
        double r150103 = 3.0;
        double r150104 = r150103 * r150092;
        double r150105 = r150100 - r150104;
        double r150106 = r150095 * r150105;
        double r150107 = r150102 + r150106;
        double r150108 = r150099 * r150107;
        double r150109 = r150098 + r150108;
        double r150110 = r150109 - r150100;
        return r150110;
}
↓
double f(double a, double b) {
        double r150111 = a;
        double r150112 = 2.0;
        double r150113 = pow(r150111, r150112);
        double r150114 = 1.0;
        double r150115 = r150114 + r150111;
        double r150116 = b;
        double r150117 = r150116 * r150116;
        double r150118 = 3.0;
        double r150119 = r150118 * r150111;
        double r150120 = r150114 - r150119;
        double r150121 = r150117 * r150120;
        double r150122 = fma(r150113, r150115, r150121);
        double r150123 = 4.0;
        double r150124 = fma(r150111, r150111, r150117);
        double r150125 = 2.0;
        double r150126 = pow(r150124, r150125);
        double r150127 = fma(r150122, r150123, r150126);
        double r150128 = sqrt(r150127);
        double r150129 = r150128 * r150128;
        double r150130 = r150129 - r150114;
        return r150130;
}

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

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020043 +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))