Average Error: 14.7 → 0.3
Time: 7.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{1}{\frac{a \cdot b}{\pi}}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{1}{\frac{a \cdot b}{\pi}}}{b + a}
double f(double a, double b) {
        double r46238 = atan2(1.0, 0.0);
        double r46239 = 2.0;
        double r46240 = r46238 / r46239;
        double r46241 = 1.0;
        double r46242 = b;
        double r46243 = r46242 * r46242;
        double r46244 = a;
        double r46245 = r46244 * r46244;
        double r46246 = r46243 - r46245;
        double r46247 = r46241 / r46246;
        double r46248 = r46240 * r46247;
        double r46249 = r46241 / r46244;
        double r46250 = r46241 / r46242;
        double r46251 = r46249 - r46250;
        double r46252 = r46248 * r46251;
        return r46252;
}
↓
double f(double a, double b) {
        double r46253 = 0.5;
        double r46254 = 1.0;
        double r46255 = a;
        double r46256 = b;
        double r46257 = r46255 * r46256;
        double r46258 = atan2(1.0, 0.0);
        double r46259 = r46257 / r46258;
        double r46260 = r46254 / r46259;
        double r46261 = r46253 * r46260;
        double r46262 = r46256 + r46255;
        double r46263 = r46261 / r46262;
        return r46263;
}

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 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.2

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

  6. Applied associate-*r*9.2

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

  8. Applied associate-*r/9.1

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

  9. Applied associate-*l/9.1

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

  10. Applied associate-*l/0.3

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

  11. Simplified0.3

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

  12. Taylor expanded around 0 0.2

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

  14. Applied clear-num0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020081 +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))))