Average Error: 14.3 → 0.3
Time: 5.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)\]
\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{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)
double f(double a, double b) {
        double r61295 = atan2(1.0, 0.0);
        double r61296 = 2.0;
        double r61297 = r61295 / r61296;
        double r61298 = 1.0;
        double r61299 = b;
        double r61300 = r61299 * r61299;
        double r61301 = a;
        double r61302 = r61301 * r61301;
        double r61303 = r61300 - r61302;
        double r61304 = r61298 / r61303;
        double r61305 = r61297 * r61304;
        double r61306 = r61298 / r61301;
        double r61307 = r61298 / r61299;
        double r61308 = r61306 - r61307;
        double r61309 = r61305 * r61308;
        return r61309;
}
↓
double f(double a, double b) {
        double r61310 = atan2(1.0, 0.0);
        double r61311 = 2.0;
        double r61312 = r61310 / r61311;
        double r61313 = b;
        double r61314 = a;
        double r61315 = r61313 + r61314;
        double r61316 = r61312 / r61315;
        double r61317 = 1.0;
        double r61318 = r61313 - r61314;
        double r61319 = r61317 / r61318;
        double r61320 = r61317 / r61314;
        double r61321 = r61317 / r61313;
        double r61322 = r61320 - r61321;
        double r61323 = r61319 * r61322;
        double r61324 = r61316 * r61323;
        return r61324;
}

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 14.3

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

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

  9. Applied associate-*l*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))