Average Error: 0.2 → 0.2
Time: 21.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \left|\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}}\right| - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \left|\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}}\right| - 1
double f(double a, double b) {
        double r10817190 = a;
        double r10817191 = r10817190 * r10817190;
        double r10817192 = b;
        double r10817193 = r10817192 * r10817192;
        double r10817194 = r10817191 + r10817193;
        double r10817195 = 2.0;
        double r10817196 = pow(r10817194, r10817195);
        double r10817197 = 4.0;
        double r10817198 = r10817197 * r10817193;
        double r10817199 = r10817196 + r10817198;
        double r10817200 = 1.0;
        double r10817201 = r10817199 - r10817200;
        return r10817201;
}
↓
double f(double a, double b) {
        double r10817202 = b;
        double r10817203 = r10817202 * r10817202;
        double r10817204 = 4.0;
        double r10817205 = r10817203 * r10817204;
        double r10817206 = a;
        double r10817207 = r10817206 * r10817206;
        double r10817208 = r10817207 + r10817203;
        double r10817209 = 2.0;
        double r10817210 = pow(r10817208, r10817209);
        double r10817211 = r10817205 + r10817210;
        double r10817212 = sqrt(r10817211);
        double r10817213 = fabs(r10817212);
        double r10817214 = r10817212 * r10817213;
        double r10817215 = 1.0;
        double r10817216 = r10817214 - r10817215;
        return r10817216;
}

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

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  5. Applied add-sqr-sqrt0.2

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

  6. Applied rem-sqrt-square0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))