Average Error: 14.9 → 0.3
Time: 19.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}
double f(double a, double b) {
        double r72204 = atan2(1.0, 0.0);
        double r72205 = 2.0;
        double r72206 = r72204 / r72205;
        double r72207 = 1.0;
        double r72208 = b;
        double r72209 = r72208 * r72208;
        double r72210 = a;
        double r72211 = r72210 * r72210;
        double r72212 = r72209 - r72211;
        double r72213 = r72207 / r72212;
        double r72214 = r72206 * r72213;
        double r72215 = r72207 / r72210;
        double r72216 = r72207 / r72208;
        double r72217 = r72215 - r72216;
        double r72218 = r72214 * r72217;
        return r72218;
}
↓
double f(double a, double b) {
        double r72219 = 0.5;
        double r72220 = atan2(1.0, 0.0);
        double r72221 = a;
        double r72222 = b;
        double r72223 = r72221 * r72222;
        double r72224 = r72220 / r72223;
        double r72225 = r72219 * r72224;
        double r72226 = 1.0;
        double r72227 = r72222 + r72221;
        double r72228 = r72226 / r72227;
        double r72229 = r72225 * r72228;
        return r72229;
}

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. Taylor expanded around 0 0.2

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

  13. Applied div-inv0.3

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

  14. Final simplification0.3

    \[\leadsto \left(0.5 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{b + a}\]

Reproduce

herbie shell --seed 2019322 +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))))