Average Error: 15.0 → 0.3
Time: 5.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi \cdot 1}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi \cdot 1}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}
double f(double a, double b) {
        double r39240 = atan2(1.0, 0.0);
        double r39241 = 2.0;
        double r39242 = r39240 / r39241;
        double r39243 = 1.0;
        double r39244 = b;
        double r39245 = r39244 * r39244;
        double r39246 = a;
        double r39247 = r39246 * r39246;
        double r39248 = r39245 - r39247;
        double r39249 = r39243 / r39248;
        double r39250 = r39242 * r39249;
        double r39251 = r39243 / r39246;
        double r39252 = r39243 / r39244;
        double r39253 = r39251 - r39252;
        double r39254 = r39250 * r39253;
        return r39254;
}
↓
double f(double a, double b) {
        double r39255 = atan2(1.0, 0.0);
        double r39256 = 1.0;
        double r39257 = r39255 * r39256;
        double r39258 = 2.0;
        double r39259 = r39257 / r39258;
        double r39260 = a;
        double r39261 = r39256 / r39260;
        double r39262 = b;
        double r39263 = r39256 / r39262;
        double r39264 = r39261 - r39263;
        double r39265 = r39262 + r39260;
        double r39266 = r39264 / r39265;
        double r39267 = r39262 - r39260;
        double r39268 = r39266 / r39267;
        double r39269 = r39259 * r39268;
        return r39269;
}

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 15.0

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.4

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

  6. Applied associate-*r/9.3

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-times0.3

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

  10. Applied associate-*l/0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 1}{2}}{1} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}}\]

  15. Simplified0.3

    \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2}} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]

  16. Final simplification0.3

    \[\leadsto \frac{\pi \cdot 1}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))