Average Error: 0.2 → 0.0
Time: 12.3s
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 r3705031 = a;
        double r3705032 = r3705031 * r3705031;
        double r3705033 = b;
        double r3705034 = r3705033 * r3705033;
        double r3705035 = r3705032 + r3705034;
        double r3705036 = 2.0;
        double r3705037 = pow(r3705035, r3705036);
        double r3705038 = 4.0;
        double r3705039 = 1.0;
        double r3705040 = r3705039 - r3705031;
        double r3705041 = r3705032 * r3705040;
        double r3705042 = 3.0;
        double r3705043 = r3705042 + r3705031;
        double r3705044 = r3705034 * r3705043;
        double r3705045 = r3705041 + r3705044;
        double r3705046 = r3705038 * r3705045;
        double r3705047 = r3705037 + r3705046;
        double r3705048 = r3705047 - r3705039;
        return r3705048;
}
↓
double f(double a, double b) {
        double r3705049 = 12.0;
        double r3705050 = b;
        double r3705051 = r3705050 * r3705050;
        double r3705052 = r3705049 * r3705051;
        double r3705053 = 4.0;
        double r3705054 = 1.0;
        double r3705055 = a;
        double r3705056 = r3705054 - r3705055;
        double r3705057 = r3705056 * r3705055;
        double r3705058 = r3705057 + r3705051;
        double r3705059 = r3705058 * r3705055;
        double r3705060 = r3705053 * r3705059;
        double r3705061 = r3705052 + r3705060;
        double r3705062 = r3705055 * r3705055;
        double r3705063 = r3705062 + r3705051;
        double r3705064 = sqrt(r3705063);
        double r3705065 = pow(r3705064, r3705053);
        double r3705066 = r3705054 - r3705065;
        double r3705067 = r3705061 - r3705066;
        return r3705067;
}

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. Using strategy rm

  10. Applied pow10.1

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

  11. Applied pow-prod-up0.0

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

  12. Simplified0.0

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

  13. 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 2019156 
(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))