Average Error: 0.2 → 0.0
Time: 6.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{\left(2 \cdot 2\right)} + 4 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(1 \cdot 1 - a \cdot a\right)}{1 - a} + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
double f(double a, double b) {
        double r123280 = a;
        double r123281 = r123280 * r123280;
        double r123282 = b;
        double r123283 = r123282 * r123282;
        double r123284 = r123281 + r123283;
        double r123285 = 2.0;
        double r123286 = pow(r123284, r123285);
        double r123287 = 4.0;
        double r123288 = 1.0;
        double r123289 = r123288 + r123280;
        double r123290 = r123281 * r123289;
        double r123291 = 3.0;
        double r123292 = r123291 * r123280;
        double r123293 = r123288 - r123292;
        double r123294 = r123283 * r123293;
        double r123295 = r123290 + r123294;
        double r123296 = r123287 * r123295;
        double r123297 = r123286 + r123296;
        double r123298 = r123297 - r123288;
        return r123298;
}
↓
double f(double a, double b) {
        double r123299 = a;
        double r123300 = b;
        double r123301 = hypot(r123299, r123300);
        double r123302 = 2.0;
        double r123303 = 2.0;
        double r123304 = r123302 * r123303;
        double r123305 = pow(r123301, r123304);
        double r123306 = 4.0;
        double r123307 = r123299 * r123299;
        double r123308 = 1.0;
        double r123309 = r123308 * r123308;
        double r123310 = r123309 - r123307;
        double r123311 = r123307 * r123310;
        double r123312 = r123308 - r123299;
        double r123313 = r123311 / r123312;
        double r123314 = r123300 * r123300;
        double r123315 = 3.0;
        double r123316 = r123315 * r123299;
        double r123317 = r123308 - r123316;
        double r123318 = r123314 * r123317;
        double r123319 = r123313 + r123318;
        double r123320 = r123306 * r123319;
        double r123321 = r123305 + r123320;
        double r123322 = r123321 - r123308;
        return r123322;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied unpow-prod-down0.2

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

  5. Applied fma-def0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}, {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\]
  6. Using strategy rm

  7. Applied fma-udef0.2

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

  8. Simplified0.0

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

  10. Applied flip-+0.0

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

  11. Applied associate-*r/0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))