Average Error: 14.2 → 0.2
Time: 19.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{a \cdot b}
double f(double a, double b) {
        double r2370304 = atan2(1.0, 0.0);
        double r2370305 = 2.0;
        double r2370306 = r2370304 / r2370305;
        double r2370307 = 1.0;
        double r2370308 = b;
        double r2370309 = r2370308 * r2370308;
        double r2370310 = a;
        double r2370311 = r2370310 * r2370310;
        double r2370312 = r2370309 - r2370311;
        double r2370313 = r2370307 / r2370312;
        double r2370314 = r2370306 * r2370313;
        double r2370315 = r2370307 / r2370310;
        double r2370316 = r2370307 / r2370308;
        double r2370317 = r2370315 - r2370316;
        double r2370318 = r2370314 * r2370317;
        return r2370318;
}
↓
double f(double a, double b) {
        double r2370319 = 1.0;
        double r2370320 = atan2(1.0, 0.0);
        double r2370321 = 2.0;
        double r2370322 = r2370320 / r2370321;
        double r2370323 = a;
        double r2370324 = b;
        double r2370325 = r2370323 + r2370324;
        double r2370326 = r2370322 / r2370325;
        double r2370327 = r2370319 * r2370326;
        double r2370328 = r2370323 * r2370324;
        double r2370329 = r2370327 / r2370328;
        return r2370329;
}

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 14.2

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

  3. Applied difference-of-squares9.4

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

  4. Applied *-un-lft-identity9.4

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*l*0.3

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

  10. Taylor expanded around 0 0.3

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

  12. Applied associate-*r/0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))