Average Error: 14.2 → 0.3
Time: 2.5m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a}}{b - a} \cdot \frac{\frac{\pi}{2}}{a + b} - \frac{\frac{\frac{\pi}{2}}{a + b}}{b \cdot \left(b - a\right)}\]
\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{1}{a}}{b - a} \cdot \frac{\frac{\pi}{2}}{a + b} - \frac{\frac{\frac{\pi}{2}}{a + b}}{b \cdot \left(b - a\right)}
double f(double a, double b) {
        double r3094729 = atan2(1.0, 0.0);
        double r3094730 = 2.0;
        double r3094731 = r3094729 / r3094730;
        double r3094732 = 1.0;
        double r3094733 = b;
        double r3094734 = r3094733 * r3094733;
        double r3094735 = a;
        double r3094736 = r3094735 * r3094735;
        double r3094737 = r3094734 - r3094736;
        double r3094738 = r3094732 / r3094737;
        double r3094739 = r3094731 * r3094738;
        double r3094740 = r3094732 / r3094735;
        double r3094741 = r3094732 / r3094733;
        double r3094742 = r3094740 - r3094741;
        double r3094743 = r3094739 * r3094742;
        return r3094743;
}
↓
double f(double a, double b) {
        double r3094744 = 1.0;
        double r3094745 = a;
        double r3094746 = r3094744 / r3094745;
        double r3094747 = b;
        double r3094748 = r3094747 - r3094745;
        double r3094749 = r3094746 / r3094748;
        double r3094750 = atan2(1.0, 0.0);
        double r3094751 = 2.0;
        double r3094752 = r3094750 / r3094751;
        double r3094753 = r3094745 + r3094747;
        double r3094754 = r3094752 / r3094753;
        double r3094755 = r3094749 * r3094754;
        double r3094756 = r3094747 * r3094748;
        double r3094757 = r3094754 / r3094756;
        double r3094758 = r3094755 - r3094757;
        return r3094758;
}

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 14.2

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

  2. Simplified9.0

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

  4. Applied associate-/l/4.5

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

  6. Applied div-inv4.5

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

  7. Applied associate-/l*0.3

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

  9. Applied div-inv0.3

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

  11. Applied associate-/r/0.3

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

  12. Applied associate-/r*0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))