Average Error: 14.9 → 0.3
Time: 6.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\pi}{2} \cdot 1\right) \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]
\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}{2} \cdot 1\right) \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}
double f(double a, double b) {
        double r47122 = atan2(1.0, 0.0);
        double r47123 = 2.0;
        double r47124 = r47122 / r47123;
        double r47125 = 1.0;
        double r47126 = b;
        double r47127 = r47126 * r47126;
        double r47128 = a;
        double r47129 = r47128 * r47128;
        double r47130 = r47127 - r47129;
        double r47131 = r47125 / r47130;
        double r47132 = r47124 * r47131;
        double r47133 = r47125 / r47128;
        double r47134 = r47125 / r47126;
        double r47135 = r47133 - r47134;
        double r47136 = r47132 * r47135;
        return r47136;
}
↓
double f(double a, double b) {
        double r47137 = atan2(1.0, 0.0);
        double r47138 = 2.0;
        double r47139 = r47137 / r47138;
        double r47140 = 1.0;
        double r47141 = r47139 * r47140;
        double r47142 = a;
        double r47143 = r47140 / r47142;
        double r47144 = b;
        double r47145 = r47140 / r47144;
        double r47146 = r47143 - r47145;
        double r47147 = r47144 + r47142;
        double r47148 = r47146 / r47147;
        double r47149 = r47141 * r47148;
        double r47150 = r47144 - r47142;
        double r47151 = r47149 / r47150;
        return r47151;
}

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 14.9

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 *-un-lft-identity9.9

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

  5. Applied times-frac9.5

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

  6. Applied associate-*r*9.5

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

  7. Simplified9.4

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

  9. Applied associate-*l/9.4

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

  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. Using strategy rm

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  16. Applied associate-*r/0.3

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

  17. Applied associate-*l/0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))