Average Error: 14.3 → 0.3
Time: 52.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\frac{\frac{-1}{\frac{a + b}{\pi}}}{2}}{b} + \frac{\pi}{\frac{a}{\frac{1}{2}} \cdot \left(a + b\right)}\right) \cdot \frac{1}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{\frac{\frac{-1}{\frac{a + b}{\pi}}}{2}}{b} + \frac{\pi}{\frac{a}{\frac{1}{2}} \cdot \left(a + b\right)}\right) \cdot \frac{1}{b - a}
double f(double a, double b) {
        double r2341581 = atan2(1.0, 0.0);
        double r2341582 = 2.0;
        double r2341583 = r2341581 / r2341582;
        double r2341584 = 1.0;
        double r2341585 = b;
        double r2341586 = r2341585 * r2341585;
        double r2341587 = a;
        double r2341588 = r2341587 * r2341587;
        double r2341589 = r2341586 - r2341588;
        double r2341590 = r2341584 / r2341589;
        double r2341591 = r2341583 * r2341590;
        double r2341592 = r2341584 / r2341587;
        double r2341593 = r2341584 / r2341585;
        double r2341594 = r2341592 - r2341593;
        double r2341595 = r2341591 * r2341594;
        return r2341595;
}
↓
double f(double a, double b) {
        double r2341596 = -1.0;
        double r2341597 = a;
        double r2341598 = b;
        double r2341599 = r2341597 + r2341598;
        double r2341600 = atan2(1.0, 0.0);
        double r2341601 = r2341599 / r2341600;
        double r2341602 = r2341596 / r2341601;
        double r2341603 = 2.0;
        double r2341604 = r2341602 / r2341603;
        double r2341605 = r2341604 / r2341598;
        double r2341606 = 0.5;
        double r2341607 = r2341597 / r2341606;
        double r2341608 = r2341607 * r2341599;
        double r2341609 = r2341600 / r2341608;
        double r2341610 = r2341605 + r2341609;
        double r2341611 = 1.0;
        double r2341612 = r2341598 - r2341597;
        double r2341613 = r2341611 / r2341612;
        double r2341614 = r2341610 * r2341613;
        return r2341614;
}

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. Simplified0.2

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

  4. Applied *-un-lft-identity0.2

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

  5. Applied div-inv0.3

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

  6. Applied times-frac0.3

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

  7. Applied associate-/l*0.3

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

  9. Applied div-inv0.3

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

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

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

  11. Applied times-frac0.3

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

  12. Simplified0.2

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

  14. Applied div-inv0.3

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

  16. Applied clear-num0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))