Average Error: 0.2 → 0.1
Time: 27.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\left(\sqrt{b \cdot b + a \cdot a} \cdot \left(\sqrt{b \cdot b + a \cdot a} \cdot \left(b \cdot b + a \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot -12\right) + \left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(\left(\sqrt{b \cdot b + a \cdot a} \cdot \left(\sqrt{b \cdot b + a \cdot a} \cdot \left(b \cdot b + a \cdot a\right)\right) + \left(\left(b \cdot b\right) \cdot a\right) \cdot -12\right) + \left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r4752271 = a;
        double r4752272 = r4752271 * r4752271;
        double r4752273 = b;
        double r4752274 = r4752273 * r4752273;
        double r4752275 = r4752272 + r4752274;
        double r4752276 = 2.0;
        double r4752277 = pow(r4752275, r4752276);
        double r4752278 = 4.0;
        double r4752279 = 1.0;
        double r4752280 = r4752279 + r4752271;
        double r4752281 = r4752272 * r4752280;
        double r4752282 = 3.0;
        double r4752283 = r4752282 * r4752271;
        double r4752284 = r4752279 - r4752283;
        double r4752285 = r4752274 * r4752284;
        double r4752286 = r4752281 + r4752285;
        double r4752287 = r4752278 * r4752286;
        double r4752288 = r4752277 + r4752287;
        double r4752289 = r4752288 - r4752279;
        return r4752289;
}
↓
double f(double a, double b) {
        double r4752290 = b;
        double r4752291 = r4752290 * r4752290;
        double r4752292 = a;
        double r4752293 = r4752292 * r4752292;
        double r4752294 = r4752291 + r4752293;
        double r4752295 = sqrt(r4752294);
        double r4752296 = r4752295 * r4752294;
        double r4752297 = r4752295 * r4752296;
        double r4752298 = r4752291 * r4752292;
        double r4752299 = -12.0;
        double r4752300 = r4752298 * r4752299;
        double r4752301 = r4752297 + r4752300;
        double r4752302 = r4752293 * r4752292;
        double r4752303 = r4752302 + r4752294;
        double r4752304 = 4.0;
        double r4752305 = r4752303 * r4752304;
        double r4752306 = r4752301 + r4752305;
        double r4752307 = 1.0;
        double r4752308 = r4752306 - r4752307;
        return r4752308;
}

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

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*l*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))