Average Error: 14.2 → 0.3
Time: 5.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \left(1 \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)\]
\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(1 \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)
double f(double a, double b) {
        double r39313 = atan2(1.0, 0.0);
        double r39314 = 2.0;
        double r39315 = r39313 / r39314;
        double r39316 = 1.0;
        double r39317 = b;
        double r39318 = r39317 * r39317;
        double r39319 = a;
        double r39320 = r39319 * r39319;
        double r39321 = r39318 - r39320;
        double r39322 = r39316 / r39321;
        double r39323 = r39315 * r39322;
        double r39324 = r39316 / r39319;
        double r39325 = r39316 / r39317;
        double r39326 = r39324 - r39325;
        double r39327 = r39323 * r39326;
        return r39327;
}
↓
double f(double a, double b) {
        double r39328 = atan2(1.0, 0.0);
        double r39329 = 2.0;
        double r39330 = r39328 / r39329;
        double r39331 = b;
        double r39332 = a;
        double r39333 = r39331 + r39332;
        double r39334 = r39330 / r39333;
        double r39335 = 1.0;
        double r39336 = r39335 / r39332;
        double r39337 = r39335 / r39331;
        double r39338 = r39336 - r39337;
        double r39339 = r39331 - r39332;
        double r39340 = r39338 / r39339;
        double r39341 = r39335 * r39340;
        double r39342 = r39334 * r39341;
        return r39342;
}

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-squares9.5

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

  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. 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. Simplified9.0

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

  11. Applied div-inv0.3

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

  12. Applied associate-*l*0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))