Average Error: 14.1 → 0.3
Time: 16.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\pi \cdot \sqrt{1}}{2 \cdot \left(b + a\right)} \cdot \sqrt{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{\pi \cdot \sqrt{1}}{2 \cdot \left(b + a\right)} \cdot \sqrt{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r59867 = atan2(1.0, 0.0);
        double r59868 = 2.0;
        double r59869 = r59867 / r59868;
        double r59870 = 1.0;
        double r59871 = b;
        double r59872 = r59871 * r59871;
        double r59873 = a;
        double r59874 = r59873 * r59873;
        double r59875 = r59872 - r59874;
        double r59876 = r59870 / r59875;
        double r59877 = r59869 * r59876;
        double r59878 = r59870 / r59873;
        double r59879 = r59870 / r59871;
        double r59880 = r59878 - r59879;
        double r59881 = r59877 * r59880;
        return r59881;
}
↓
double f(double a, double b) {
        double r59882 = atan2(1.0, 0.0);
        double r59883 = 1.0;
        double r59884 = sqrt(r59883);
        double r59885 = r59882 * r59884;
        double r59886 = 2.0;
        double r59887 = b;
        double r59888 = a;
        double r59889 = r59887 + r59888;
        double r59890 = r59886 * r59889;
        double r59891 = r59885 / r59890;
        double r59892 = r59891 * r59884;
        double r59893 = r59883 / r59888;
        double r59894 = r59883 / r59887;
        double r59895 = r59893 - r59894;
        double r59896 = r59892 * r59895;
        double r59897 = r59887 - r59888;
        double r59898 = r59896 / r59897;
        return r59898;
}

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-squares9.6

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

  4. Applied add-sqr-sqrt9.6

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  8. Applied associate-*r/9.0

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

  9. Applied associate-*l/0.3

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

  11. Applied frac-times0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))