Average Error: 14.5 → 0.8
Time: 25.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}
double f(double a, double b) {
        double r54629 = atan2(1.0, 0.0);
        double r54630 = 2.0;
        double r54631 = r54629 / r54630;
        double r54632 = 1.0;
        double r54633 = b;
        double r54634 = r54633 * r54633;
        double r54635 = a;
        double r54636 = r54635 * r54635;
        double r54637 = r54634 - r54636;
        double r54638 = r54632 / r54637;
        double r54639 = r54631 * r54638;
        double r54640 = r54632 / r54635;
        double r54641 = r54632 / r54633;
        double r54642 = r54640 - r54641;
        double r54643 = r54639 * r54642;
        return r54643;
}
↓
double f(double a, double b) {
        double r54644 = 1.0;
        double r54645 = b;
        double r54646 = a;
        double r54647 = r54645 - r54646;
        double r54648 = r54644 * r54647;
        double r54649 = atan2(1.0, 0.0);
        double r54650 = 2.0;
        double r54651 = r54649 / r54650;
        double r54652 = r54645 + r54646;
        double r54653 = r54651 / r54652;
        double r54654 = r54653 * r54644;
        double r54655 = r54648 * r54654;
        double r54656 = expm1(r54655);
        double r54657 = log1p(r54656);
        double r54658 = r54646 * r54645;
        double r54659 = r54647 * r54658;
        double r54660 = r54657 / r54659;
        return r54660;
}

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 14.5

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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. Simplified9.3

    \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{2}}{b + a}} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 frac-sub0.3

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

  13. Applied associate-*r/0.3

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

  14. Applied associate-/l/0.8

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

  16. Applied log1p-expm1-u0.8

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right)\right)}}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

  17. Simplified0.8

    \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)}\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

  18. Final simplification0.8

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

Reproduce

herbie shell --seed 2019325 +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))))