Average Error: 13.7 → 0.2
Time: 54.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a \cdot b} \cdot 0.5}{a + b}\]
\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{\pi}{a \cdot b} \cdot 0.5}{a + b}
double f(double a, double b) {
        double r2829076 = atan2(1.0, 0.0);
        double r2829077 = 2.0;
        double r2829078 = r2829076 / r2829077;
        double r2829079 = 1.0;
        double r2829080 = b;
        double r2829081 = r2829080 * r2829080;
        double r2829082 = a;
        double r2829083 = r2829082 * r2829082;
        double r2829084 = r2829081 - r2829083;
        double r2829085 = r2829079 / r2829084;
        double r2829086 = r2829078 * r2829085;
        double r2829087 = r2829079 / r2829082;
        double r2829088 = r2829079 / r2829080;
        double r2829089 = r2829087 - r2829088;
        double r2829090 = r2829086 * r2829089;
        return r2829090;
}
↓
double f(double a, double b) {
        double r2829091 = atan2(1.0, 0.0);
        double r2829092 = a;
        double r2829093 = b;
        double r2829094 = r2829092 * r2829093;
        double r2829095 = r2829091 / r2829094;
        double r2829096 = 0.5;
        double r2829097 = r2829095 * r2829096;
        double r2829098 = r2829092 + r2829093;
        double r2829099 = r2829097 / r2829098;
        return r2829099;
}

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 13.7

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +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))))