Average Error: 14.1 → 0.2
Time: 36.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{a + b}\]
\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}}{a + b}
double f(double a, double b) {
        double r2757945 = atan2(1.0, 0.0);
        double r2757946 = 2.0;
        double r2757947 = r2757945 / r2757946;
        double r2757948 = 1.0;
        double r2757949 = b;
        double r2757950 = r2757949 * r2757949;
        double r2757951 = a;
        double r2757952 = r2757951 * r2757951;
        double r2757953 = r2757950 - r2757952;
        double r2757954 = r2757948 / r2757953;
        double r2757955 = r2757947 * r2757954;
        double r2757956 = r2757948 / r2757951;
        double r2757957 = r2757948 / r2757949;
        double r2757958 = r2757956 - r2757957;
        double r2757959 = r2757955 * r2757958;
        return r2757959;
}
↓
double f(double a, double b) {
        double r2757960 = 0.5;
        double r2757961 = atan2(1.0, 0.0);
        double r2757962 = r2757960 * r2757961;
        double r2757963 = a;
        double r2757964 = b;
        double r2757965 = r2757963 * r2757964;
        double r2757966 = r2757962 / r2757965;
        double r2757967 = r2757963 + r2757964;
        double r2757968 = r2757966 / r2757967;
        return r2757968;
}

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 14.1

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

  3. Applied difference-of-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.7

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

  9. Applied associate-*l/8.7

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

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

  13. Applied associate-*r/0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))