Average Error: 14.3 → 0.2
Time: 6.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b + a}
double f(double a, double b) {
        double r43010 = atan2(1.0, 0.0);
        double r43011 = 2.0;
        double r43012 = r43010 / r43011;
        double r43013 = 1.0;
        double r43014 = b;
        double r43015 = r43014 * r43014;
        double r43016 = a;
        double r43017 = r43016 * r43016;
        double r43018 = r43015 - r43017;
        double r43019 = r43013 / r43018;
        double r43020 = r43012 * r43019;
        double r43021 = r43013 / r43016;
        double r43022 = r43013 / r43014;
        double r43023 = r43021 - r43022;
        double r43024 = r43020 * r43023;
        return r43024;
}
↓
double f(double a, double b) {
        double r43025 = 0.5;
        double r43026 = atan2(1.0, 0.0);
        double r43027 = a;
        double r43028 = b;
        double r43029 = r43027 * r43028;
        double r43030 = r43026 / r43029;
        double r43031 = r43025 * r43030;
        double r43032 = r43028 + r43027;
        double r43033 = r43031 / r43032;
        return r43033;
}

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 14.3

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  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. Simplified9.0

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

  9. Applied associate-*l/9.0

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020027 +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))))