Average Error: 14.7 → 0.3
Time: 8.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\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{\frac{\pi}{2}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r54622 = atan2(1.0, 0.0);
        double r54623 = 2.0;
        double r54624 = r54622 / r54623;
        double r54625 = 1.0;
        double r54626 = b;
        double r54627 = r54626 * r54626;
        double r54628 = a;
        double r54629 = r54628 * r54628;
        double r54630 = r54627 - r54629;
        double r54631 = r54625 / r54630;
        double r54632 = r54624 * r54631;
        double r54633 = r54625 / r54628;
        double r54634 = r54625 / r54626;
        double r54635 = r54633 - r54634;
        double r54636 = r54632 * r54635;
        return r54636;
}
↓
double f(double a, double b) {
        double r54637 = atan2(1.0, 0.0);
        double r54638 = 2.0;
        double r54639 = r54637 / r54638;
        double r54640 = b;
        double r54641 = a;
        double r54642 = r54640 + r54641;
        double r54643 = r54639 / r54642;
        double r54644 = 1.0;
        double r54645 = r54643 * r54644;
        double r54646 = r54640 - r54641;
        double r54647 = r54644 / r54641;
        double r54648 = r54644 / r54640;
        double r54649 = r54647 - r54648;
        double r54650 = r54646 / r54649;
        double r54651 = r54645 / r54650;
        return r54651;
}

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 14.7

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

  3. Applied difference-of-squares9.8

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

  4. Applied *-un-lft-identity9.8

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

  5. Applied times-frac9.3

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

  6. Applied associate-*r*9.3

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

  7. Simplified9.2

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

  9. Applied associate-*r/9.2

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))