Average Error: 14.7 → 0.2
Time: 6.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{b + a}\]
\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{0.5 \cdot \pi}{a \cdot b}}{b + a}
double f(double a, double b) {
        double r46092 = atan2(1.0, 0.0);
        double r46093 = 2.0;
        double r46094 = r46092 / r46093;
        double r46095 = 1.0;
        double r46096 = b;
        double r46097 = r46096 * r46096;
        double r46098 = a;
        double r46099 = r46098 * r46098;
        double r46100 = r46097 - r46099;
        double r46101 = r46095 / r46100;
        double r46102 = r46094 * r46101;
        double r46103 = r46095 / r46098;
        double r46104 = r46095 / r46096;
        double r46105 = r46103 - r46104;
        double r46106 = r46102 * r46105;
        return r46106;
}
↓
double f(double a, double b) {
        double r46107 = 0.5;
        double r46108 = atan2(1.0, 0.0);
        double r46109 = r46107 * r46108;
        double r46110 = a;
        double r46111 = b;
        double r46112 = r46110 * r46111;
        double r46113 = r46109 / r46112;
        double r46114 = r46111 + r46110;
        double r46115 = r46113 / r46114;
        return r46115;
}

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-squares9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.1

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

  9. Applied associate-*l/9.1

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

  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 associate-*r/0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020083 +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))))