Average Error: 14.0 → 0.3
Time: 11.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{2 \cdot \left(b - a\right)}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{2 \cdot \left(b - a\right)}}{b + a}
double f(double a, double b) {
        double r40233 = atan2(1.0, 0.0);
        double r40234 = 2.0;
        double r40235 = r40233 / r40234;
        double r40236 = 1.0;
        double r40237 = b;
        double r40238 = r40237 * r40237;
        double r40239 = a;
        double r40240 = r40239 * r40239;
        double r40241 = r40238 - r40240;
        double r40242 = r40236 / r40241;
        double r40243 = r40235 * r40242;
        double r40244 = r40236 / r40239;
        double r40245 = r40236 / r40237;
        double r40246 = r40244 - r40245;
        double r40247 = r40243 * r40246;
        return r40247;
}
↓
double f(double a, double b) {
        double r40248 = 1.0;
        double r40249 = atan2(1.0, 0.0);
        double r40250 = a;
        double r40251 = r40249 / r40250;
        double r40252 = b;
        double r40253 = r40249 / r40252;
        double r40254 = r40251 - r40253;
        double r40255 = r40248 * r40254;
        double r40256 = 2.0;
        double r40257 = r40252 - r40250;
        double r40258 = r40256 * r40257;
        double r40259 = r40255 / r40258;
        double r40260 = r40252 + r40250;
        double r40261 = r40259 / r40260;
        return r40261;
}

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 14.0

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  7. Applied associate-*l/9.0

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

  8. Applied associate-*r/9.0

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

  9. Applied associate-*l/0.3

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

  10. Simplified0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied frac-times0.3

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

  14. Taylor expanded around 0 0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020043 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))