Average Error: 14.0 → 0.3
Time: 10.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5}{a} \cdot \frac{\pi}{b}}{b + a}\]
\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{0.5}{a} \cdot \frac{\pi}{b}}{b + a}
double f(double a, double b) {
        double r53211 = atan2(1.0, 0.0);
        double r53212 = 2.0;
        double r53213 = r53211 / r53212;
        double r53214 = 1.0;
        double r53215 = b;
        double r53216 = r53215 * r53215;
        double r53217 = a;
        double r53218 = r53217 * r53217;
        double r53219 = r53216 - r53218;
        double r53220 = r53214 / r53219;
        double r53221 = r53213 * r53220;
        double r53222 = r53214 / r53217;
        double r53223 = r53214 / r53215;
        double r53224 = r53222 - r53223;
        double r53225 = r53221 * r53224;
        return r53225;
}
↓
double f(double a, double b) {
        double r53226 = 0.5;
        double r53227 = a;
        double r53228 = r53226 / r53227;
        double r53229 = atan2(1.0, 0.0);
        double r53230 = b;
        double r53231 = r53229 / r53230;
        double r53232 = r53228 * r53231;
        double r53233 = r53230 + r53227;
        double r53234 = r53232 / r53233;
        return r53234;
}

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-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.9

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

  7. Simplified8.8

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

  9. Applied associate-*l/8.8

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity0.2

    \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{1 \cdot \pi}}{a \cdot b}}{b + a}\]

  14. Applied times-frac0.3

    \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{\pi}{b}\right)}}{b + a}\]

  15. Applied associate-*r*0.3

    \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \frac{1}{a}\right) \cdot \frac{\pi}{b}}}{b + a}\]

  16. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{0.5}{a}} \cdot \frac{\pi}{b}}{b + a}\]

  17. Final simplification0.3

    \[\leadsto \frac{\frac{0.5}{a} \cdot \frac{\pi}{b}}{b + a}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))