Average Error: 0.2 → 0.0
Time: 1.7m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r82936197 = a;
        double r82936198 = r82936197 * r82936197;
        double r82936199 = b;
        double r82936200 = r82936199 * r82936199;
        double r82936201 = r82936198 + r82936200;
        double r82936202 = 2.0;
        double r82936203 = pow(r82936201, r82936202);
        double r82936204 = 4.0;
        double r82936205 = r82936204 * r82936200;
        double r82936206 = r82936203 + r82936205;
        double r82936207 = 1.0;
        double r82936208 = r82936206 - r82936207;
        return r82936208;
}
↓
double f(double a, double b) {
        double r82936209 = 4.0;
        double r82936210 = b;
        double r82936211 = r82936210 * r82936210;
        double r82936212 = r82936209 * r82936211;
        double r82936213 = a;
        double r82936214 = r82936213 * r82936213;
        double r82936215 = r82936211 + r82936214;
        double r82936216 = sqrt(r82936215);
        double r82936217 = pow(r82936216, r82936209);
        double r82936218 = r82936212 + r82936217;
        double r82936219 = 1.0;
        double r82936220 = r82936218 - r82936219;
        return r82936220;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied cube-unmult0.1

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

  9. Applied pow10.1

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

  10. Applied pow-prod-up0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))