Average Error: 0.2 → 0.2
Time: 5.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{\sqrt[3]{1 + a} \cdot \sqrt[3]{1 + a}}}{\sqrt[3]{1 + a}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\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({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{\sqrt[3]{1 + a} \cdot \sqrt[3]{1 + a}}}{\sqrt[3]{1 + a}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
double f(double a, double b) {
        double r237027 = a;
        double r237028 = r237027 * r237027;
        double r237029 = b;
        double r237030 = r237029 * r237029;
        double r237031 = r237028 + r237030;
        double r237032 = 2.0;
        double r237033 = pow(r237031, r237032);
        double r237034 = 4.0;
        double r237035 = 1.0;
        double r237036 = r237035 - r237027;
        double r237037 = r237028 * r237036;
        double r237038 = 3.0;
        double r237039 = r237038 + r237027;
        double r237040 = r237030 * r237039;
        double r237041 = r237037 + r237040;
        double r237042 = r237034 * r237041;
        double r237043 = r237033 + r237042;
        double r237044 = r237043 - r237035;
        return r237044;
}
↓
double f(double a, double b) {
        double r237045 = a;
        double r237046 = r237045 * r237045;
        double r237047 = b;
        double r237048 = r237047 * r237047;
        double r237049 = r237046 + r237048;
        double r237050 = 2.0;
        double r237051 = pow(r237049, r237050);
        double r237052 = 4.0;
        double r237053 = 1.0;
        double r237054 = r237053 * r237053;
        double r237055 = r237054 - r237046;
        double r237056 = r237046 * r237055;
        double r237057 = r237053 + r237045;
        double r237058 = cbrt(r237057);
        double r237059 = r237058 * r237058;
        double r237060 = r237056 / r237059;
        double r237061 = r237060 / r237058;
        double r237062 = 3.0;
        double r237063 = r237062 + r237045;
        double r237064 = r237048 * r237063;
        double r237065 = r237061 + r237064;
        double r237066 = r237052 * r237065;
        double r237067 = r237051 + r237066;
        double r237068 = r237067 - r237053;
        return r237068;
}

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. Using strategy rm

  3. Applied flip--0.2

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

  4. Applied associate-*r/0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 + a}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{\color{blue}{\left(\sqrt[3]{1 + a} \cdot \sqrt[3]{1 + a}\right) \cdot \sqrt[3]{1 + a}}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  7. Applied associate-/r*0.2

    \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\frac{\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{\sqrt[3]{1 + a} \cdot \sqrt[3]{1 + a}}}{\sqrt[3]{1 + a}}} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))