Average Error: 14.3 → 0.2
Time: 5.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}\]
\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 \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r41031 = atan2(1.0, 0.0);
        double r41032 = 2.0;
        double r41033 = r41031 / r41032;
        double r41034 = 1.0;
        double r41035 = b;
        double r41036 = r41035 * r41035;
        double r41037 = a;
        double r41038 = r41037 * r41037;
        double r41039 = r41036 - r41038;
        double r41040 = r41034 / r41039;
        double r41041 = r41033 * r41040;
        double r41042 = r41034 / r41037;
        double r41043 = r41034 / r41035;
        double r41044 = r41042 - r41043;
        double r41045 = r41041 * r41044;
        return r41045;
}
↓
double f(double a, double b) {
        double r41046 = 0.5;
        double r41047 = atan2(1.0, 0.0);
        double r41048 = a;
        double r41049 = r41047 / r41048;
        double r41050 = b;
        double r41051 = r41047 / r41050;
        double r41052 = r41049 - r41051;
        double r41053 = r41046 * r41052;
        double r41054 = r41050 + r41048;
        double r41055 = r41053 / r41054;
        double r41056 = r41050 - r41048;
        double r41057 = r41055 / r41056;
        return r41057;
}

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-squares9.3

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

  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-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*r/8.9

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied associate-*l/0.3

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

  14. Taylor expanded around 0 0.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020024 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))