Average Error: 13.9 → 0.3
Time: 6.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r50266 = atan2(1.0, 0.0);
        double r50267 = 2.0;
        double r50268 = r50266 / r50267;
        double r50269 = 1.0;
        double r50270 = b;
        double r50271 = r50270 * r50270;
        double r50272 = a;
        double r50273 = r50272 * r50272;
        double r50274 = r50271 - r50273;
        double r50275 = r50269 / r50274;
        double r50276 = r50268 * r50275;
        double r50277 = r50269 / r50272;
        double r50278 = r50269 / r50270;
        double r50279 = r50277 - r50278;
        double r50280 = r50276 * r50279;
        return r50280;
}
↓
double f(double a, double b) {
        double r50281 = 1.0;
        double r50282 = b;
        double r50283 = a;
        double r50284 = r50282 + r50283;
        double r50285 = atan2(1.0, 0.0);
        double r50286 = 2.0;
        double r50287 = r50285 / r50286;
        double r50288 = r50284 / r50287;
        double r50289 = r50281 / r50288;
        double r50290 = 1.0;
        double r50291 = r50289 * r50290;
        double r50292 = r50282 - r50283;
        double r50293 = r50290 / r50283;
        double r50294 = r50290 / r50282;
        double r50295 = r50293 - r50294;
        double r50296 = r50292 / r50295;
        double r50297 = r50291 / r50296;
        return r50297;
}

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 13.9

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

  3. Applied difference-of-squares8.9

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

  4. Applied *-un-lft-identity8.9

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

  5. Applied times-frac8.6

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

  6. Applied associate-*r*8.6

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

  7. Simplified8.5

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

  9. Applied associate-*r/8.5

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

  10. Applied associate-*l/0.3

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

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

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

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

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

  14. Applied times-frac0.3

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

  15. Applied associate-/l*0.3

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

  17. Applied associate-/l*0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))