Average Error: 14.5 → 0.3
Time: 5.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot \left(1 \cdot \left(b - a\right)\right)\right) \cdot 1}{a \cdot b}}{b - a}\]
\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{\left(\frac{\frac{\pi}{2}}{b + a} \cdot \left(1 \cdot \left(b - a\right)\right)\right) \cdot 1}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r45254 = atan2(1.0, 0.0);
        double r45255 = 2.0;
        double r45256 = r45254 / r45255;
        double r45257 = 1.0;
        double r45258 = b;
        double r45259 = r45258 * r45258;
        double r45260 = a;
        double r45261 = r45260 * r45260;
        double r45262 = r45259 - r45261;
        double r45263 = r45257 / r45262;
        double r45264 = r45256 * r45263;
        double r45265 = r45257 / r45260;
        double r45266 = r45257 / r45258;
        double r45267 = r45265 - r45266;
        double r45268 = r45264 * r45267;
        return r45268;
}
↓
double f(double a, double b) {
        double r45269 = atan2(1.0, 0.0);
        double r45270 = 2.0;
        double r45271 = r45269 / r45270;
        double r45272 = b;
        double r45273 = a;
        double r45274 = r45272 + r45273;
        double r45275 = r45271 / r45274;
        double r45276 = 1.0;
        double r45277 = r45272 - r45273;
        double r45278 = r45276 * r45277;
        double r45279 = r45275 * r45278;
        double r45280 = r45279 * r45276;
        double r45281 = r45273 * r45272;
        double r45282 = r45280 / r45281;
        double r45283 = r45282 / r45277;
        return r45283;
}

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 14.5

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

  3. Applied difference-of-squares9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac9.3

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

  6. Applied associate-*r*9.2

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

  7. Simplified9.2

    \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{2}}{b + a}} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 associate-*l/0.3

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

  12. Applied associate-*r/0.3

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

  14. Applied frac-sub0.3

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

  15. Applied associate-*r/0.3

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

  16. Applied associate-*r/0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))