Average Error: 0.2 → 0.2
Time: 25.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} - 1\]
\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 \sqrt{\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}} - 1
double f(double a, double b) {
        double r5900998 = a;
        double r5900999 = r5900998 * r5900998;
        double r5901000 = b;
        double r5901001 = r5901000 * r5901000;
        double r5901002 = r5900999 + r5901001;
        double r5901003 = 2.0;
        double r5901004 = pow(r5901002, r5901003);
        double r5901005 = 4.0;
        double r5901006 = r5901005 * r5901001;
        double r5901007 = r5901004 + r5901006;
        double r5901008 = 1.0;
        double r5901009 = r5901007 - r5901008;
        return r5901009;
}
↓
double f(double a, double b) {
        double r5901010 = b;
        double r5901011 = r5901010 * r5901010;
        double r5901012 = 4.0;
        double r5901013 = r5901011 * r5901012;
        double r5901014 = a;
        double r5901015 = r5901014 * r5901014;
        double r5901016 = r5901015 + r5901011;
        double r5901017 = 2.0;
        double r5901018 = pow(r5901016, r5901017);
        double r5901019 = r5901013 + r5901018;
        double r5901020 = sqrt(r5901019);
        double r5901021 = r5901020 * r5901020;
        double r5901022 = 1.0;
        double r5901023 = r5901021 - r5901022;
        return r5901023;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))