Average Error: 14.1 → 0.3
Time: 8.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}
double f(double a, double b) {
        double r57023 = atan2(1.0, 0.0);
        double r57024 = 2.0;
        double r57025 = r57023 / r57024;
        double r57026 = 1.0;
        double r57027 = b;
        double r57028 = r57027 * r57027;
        double r57029 = a;
        double r57030 = r57029 * r57029;
        double r57031 = r57028 - r57030;
        double r57032 = r57026 / r57031;
        double r57033 = r57025 * r57032;
        double r57034 = r57026 / r57029;
        double r57035 = r57026 / r57027;
        double r57036 = r57034 - r57035;
        double r57037 = r57033 * r57036;
        return r57037;
}
↓
double f(double a, double b) {
        double r57038 = atan2(1.0, 0.0);
        double r57039 = 2.0;
        double r57040 = r57038 / r57039;
        double r57041 = b;
        double r57042 = a;
        double r57043 = r57041 + r57042;
        double r57044 = r57040 / r57043;
        double r57045 = 1.0;
        double r57046 = r57044 * r57045;
        double r57047 = r57045 / r57042;
        double r57048 = r57045 / r57041;
        double r57049 = r57047 - r57048;
        double r57050 = r57041 - r57042;
        double r57051 = r57049 / r57050;
        double r57052 = r57046 * r57051;
        return r57052;
}

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 14.1

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.7

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

  9. Applied associate-*r/8.7

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

  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

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020018 +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))))