Average Error: 0.2 → 0.0
Time: 19.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(12 \cdot \left(b \cdot b\right) + 4 \cdot \left(\left(\left(1 - a\right) \cdot a + b \cdot b\right) \cdot a\right)\right) - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]
\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(12 \cdot \left(b \cdot b\right) + 4 \cdot \left(\left(\left(1 - a\right) \cdot a + b \cdot b\right) \cdot a\right)\right) - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)
double f(double a, double b) {
        double r5158121 = a;
        double r5158122 = r5158121 * r5158121;
        double r5158123 = b;
        double r5158124 = r5158123 * r5158123;
        double r5158125 = r5158122 + r5158124;
        double r5158126 = 2.0;
        double r5158127 = pow(r5158125, r5158126);
        double r5158128 = 4.0;
        double r5158129 = 1.0;
        double r5158130 = r5158129 - r5158121;
        double r5158131 = r5158122 * r5158130;
        double r5158132 = 3.0;
        double r5158133 = r5158132 + r5158121;
        double r5158134 = r5158124 * r5158133;
        double r5158135 = r5158131 + r5158134;
        double r5158136 = r5158128 * r5158135;
        double r5158137 = r5158127 + r5158136;
        double r5158138 = r5158137 - r5158129;
        return r5158138;
}
↓
double f(double a, double b) {
        double r5158139 = 12.0;
        double r5158140 = b;
        double r5158141 = r5158140 * r5158140;
        double r5158142 = r5158139 * r5158141;
        double r5158143 = 4.0;
        double r5158144 = 1.0;
        double r5158145 = a;
        double r5158146 = r5158144 - r5158145;
        double r5158147 = r5158146 * r5158145;
        double r5158148 = r5158147 + r5158141;
        double r5158149 = r5158148 * r5158145;
        double r5158150 = r5158143 * r5158149;
        double r5158151 = r5158142 + r5158150;
        double r5158152 = r5158145 * r5158145;
        double r5158153 = r5158152 + r5158141;
        double r5158154 = sqrt(r5158153);
        double r5158155 = pow(r5158154, r5158143);
        double r5158156 = r5158144 - r5158155;
        double r5158157 = r5158151 - r5158156;
        return r5158157;
}

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

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

    \[\leadsto \left(12 \cdot \left(b \cdot b\right) + 4 \cdot \left(a \cdot \left(b \cdot b + a \cdot \left(1 - a\right)\right)\right)\right) - \left(1 - \color{blue}{\sqrt{a \cdot a + b \cdot b} \cdot \left(\sqrt{a \cdot a + b \cdot b} \cdot \left(a \cdot a + b \cdot b\right)\right)}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

    \[\leadsto \left(12 \cdot \left(b \cdot b\right) + 4 \cdot \left(\left(\left(1 - a\right) \cdot a + b \cdot b\right) \cdot a\right)\right) - \left(1 - {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))