Average Error: 15.0 → 0.3
Time: 2.0m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a + b}}{\frac{b - a}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}\]
\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 + b}}{\frac{b - a}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}
double f(double a, double b) {
        double r2731498 = atan2(1.0, 0.0);
        double r2731499 = 2.0;
        double r2731500 = r2731498 / r2731499;
        double r2731501 = 1.0;
        double r2731502 = b;
        double r2731503 = r2731502 * r2731502;
        double r2731504 = a;
        double r2731505 = r2731504 * r2731504;
        double r2731506 = r2731503 - r2731505;
        double r2731507 = r2731501 / r2731506;
        double r2731508 = r2731500 * r2731507;
        double r2731509 = r2731501 / r2731504;
        double r2731510 = r2731501 / r2731502;
        double r2731511 = r2731509 - r2731510;
        double r2731512 = r2731508 * r2731511;
        return r2731512;
}
↓
double f(double a, double b) {
        double r2731513 = atan2(1.0, 0.0);
        double r2731514 = a;
        double r2731515 = b;
        double r2731516 = r2731514 + r2731515;
        double r2731517 = r2731513 / r2731516;
        double r2731518 = r2731515 - r2731514;
        double r2731519 = 0.5;
        double r2731520 = r2731519 / r2731514;
        double r2731521 = r2731519 / r2731515;
        double r2731522 = r2731520 - r2731521;
        double r2731523 = r2731518 / r2731522;
        double r2731524 = r2731517 / r2731523;
        return r2731524;
}

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 15.0

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

  2. Simplified10.0

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

  4. Applied associate-/r*0.3

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

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

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

  7. Applied div-inv0.3

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

  8. Applied times-frac0.3

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

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

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

  10. Applied div-inv0.3

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

  11. Applied times-frac0.3

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

  12. Applied distribute-lft-out--0.3

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

  13. Applied associate-/l*0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{1}}{\frac{a + b}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}}}{b - a}\]

  14. Simplified0.3

    \[\leadsto \frac{\frac{\frac{\pi}{1}}{\color{blue}{\frac{a + b}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}}}{b - a}\]
  15. Using strategy rm

  16. Applied div-inv0.3

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

  17. Applied div-inv0.3

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

  18. Applied times-frac0.3

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

  19. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{\frac{b - a}{\frac{\frac{1}{1}}{\frac{1}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}}}}\]

  20. Simplified0.3

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

  21. Final simplification0.3

    \[\leadsto \frac{\frac{\pi}{a + b}}{\frac{b - a}{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))