Average Error: 14.4 → 0.3
Time: 1.5m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \left(\frac{1}{\left(b + a\right) \cdot a} \cdot \frac{1}{b - a}\right) - \frac{\frac{1}{\frac{b - a}{\frac{\pi}{2}}}}{\left(b + a\right) \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \left(\frac{1}{\left(b + a\right) \cdot a} \cdot \frac{1}{b - a}\right) - \frac{\frac{1}{\frac{b - a}{\frac{\pi}{2}}}}{\left(b + a\right) \cdot b}
double f(double a, double b) {
        double r3034012 = atan2(1.0, 0.0);
        double r3034013 = 2.0;
        double r3034014 = r3034012 / r3034013;
        double r3034015 = 1.0;
        double r3034016 = b;
        double r3034017 = r3034016 * r3034016;
        double r3034018 = a;
        double r3034019 = r3034018 * r3034018;
        double r3034020 = r3034017 - r3034019;
        double r3034021 = r3034015 / r3034020;
        double r3034022 = r3034014 * r3034021;
        double r3034023 = r3034015 / r3034018;
        double r3034024 = r3034015 / r3034016;
        double r3034025 = r3034023 - r3034024;
        double r3034026 = r3034022 * r3034025;
        return r3034026;
}
↓
double f(double a, double b) {
        double r3034027 = atan2(1.0, 0.0);
        double r3034028 = 2.0;
        double r3034029 = r3034027 / r3034028;
        double r3034030 = 1.0;
        double r3034031 = b;
        double r3034032 = a;
        double r3034033 = r3034031 + r3034032;
        double r3034034 = r3034033 * r3034032;
        double r3034035 = r3034030 / r3034034;
        double r3034036 = r3034031 - r3034032;
        double r3034037 = r3034030 / r3034036;
        double r3034038 = r3034035 * r3034037;
        double r3034039 = r3034029 * r3034038;
        double r3034040 = r3034036 / r3034029;
        double r3034041 = r3034030 / r3034040;
        double r3034042 = r3034033 * r3034031;
        double r3034043 = r3034041 / r3034042;
        double r3034044 = r3034039 - r3034043;
        return r3034044;
}

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 14.4

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

  2. Simplified8.7

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{2}}{b - a}}{a + b}}{a} - \frac{\frac{\frac{\frac{\pi}{2}}{b - a}}{a + b}}{b}}\]
  3. Using strategy rm

  4. Applied associate-/l/4.4

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

  6. Applied *-un-lft-identity4.4

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

  7. Applied div-inv4.4

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

  8. Applied times-frac0.3

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

  10. Applied *-un-lft-identity0.3

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

  11. Applied associate-/l*0.3

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

  13. Applied *-un-lft-identity0.3

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

  14. Applied div-inv0.3

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

  15. Applied times-frac0.3

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

  16. Applied associate-*l*0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))