Average Error: 14.1 → 0.5
Time: 21.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\left(\left(b - a\right) \cdot 1\right) \cdot \left(\pi \cdot 1\right)}{b + a}}{b \cdot a}}{2 \cdot \left(b - a\right)}\]
\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{\frac{\left(\left(b - a\right) \cdot 1\right) \cdot \left(\pi \cdot 1\right)}{b + a}}{b \cdot a}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r2560043 = atan2(1.0, 0.0);
        double r2560044 = 2.0;
        double r2560045 = r2560043 / r2560044;
        double r2560046 = 1.0;
        double r2560047 = b;
        double r2560048 = r2560047 * r2560047;
        double r2560049 = a;
        double r2560050 = r2560049 * r2560049;
        double r2560051 = r2560048 - r2560050;
        double r2560052 = r2560046 / r2560051;
        double r2560053 = r2560045 * r2560052;
        double r2560054 = r2560046 / r2560049;
        double r2560055 = r2560046 / r2560047;
        double r2560056 = r2560054 - r2560055;
        double r2560057 = r2560053 * r2560056;
        return r2560057;
}
↓
double f(double a, double b) {
        double r2560058 = b;
        double r2560059 = a;
        double r2560060 = r2560058 - r2560059;
        double r2560061 = 1.0;
        double r2560062 = r2560060 * r2560061;
        double r2560063 = atan2(1.0, 0.0);
        double r2560064 = r2560063 * r2560061;
        double r2560065 = r2560062 * r2560064;
        double r2560066 = r2560058 + r2560059;
        double r2560067 = r2560065 / r2560066;
        double r2560068 = r2560058 * r2560059;
        double r2560069 = r2560067 / r2560068;
        double r2560070 = 2.0;
        double r2560071 = r2560070 * r2560060;
        double r2560072 = r2560069 / r2560071;
        return r2560072;
}

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-squares9.2

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

  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. Simplified8.8

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

  9. Applied frac-times8.7

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

  10. Applied associate-*l/0.3

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

  12. Applied frac-sub0.4

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

  13. Applied associate-*r/0.3

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(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))))