Average Error: 15.0 → 0.2
Time: 8.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{2 \cdot \left(b + a\right)}}{b - a}\]
\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{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{2 \cdot \left(b + a\right)}}{b - a}
double f(double a, double b) {
        double r69886 = atan2(1.0, 0.0);
        double r69887 = 2.0;
        double r69888 = r69886 / r69887;
        double r69889 = 1.0;
        double r69890 = b;
        double r69891 = r69890 * r69890;
        double r69892 = a;
        double r69893 = r69892 * r69892;
        double r69894 = r69891 - r69893;
        double r69895 = r69889 / r69894;
        double r69896 = r69888 * r69895;
        double r69897 = r69889 / r69892;
        double r69898 = r69889 / r69890;
        double r69899 = r69897 - r69898;
        double r69900 = r69896 * r69899;
        return r69900;
}
↓
double f(double a, double b) {
        double r69901 = 1.0;
        double r69902 = atan2(1.0, 0.0);
        double r69903 = a;
        double r69904 = r69902 / r69903;
        double r69905 = b;
        double r69906 = r69902 / r69905;
        double r69907 = r69904 - r69906;
        double r69908 = r69901 * r69907;
        double r69909 = 2.0;
        double r69910 = r69905 + r69903;
        double r69911 = r69909 * r69910;
        double r69912 = r69908 / r69911;
        double r69913 = r69905 - r69903;
        double r69914 = r69912 / r69913;
        return r69914;
}

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 15.0

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

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.4

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

  6. Applied associate-*r/9.3

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-times0.3

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 +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))))