Average Error: 14.5 → 0.8
Time: 25.0s
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 r54738 = atan2(1.0, 0.0);
        double r54739 = 2.0;
        double r54740 = r54738 / r54739;
        double r54741 = 1.0;
        double r54742 = b;
        double r54743 = r54742 * r54742;
        double r54744 = a;
        double r54745 = r54744 * r54744;
        double r54746 = r54743 - r54745;
        double r54747 = r54741 / r54746;
        double r54748 = r54740 * r54747;
        double r54749 = r54741 / r54744;
        double r54750 = r54741 / r54742;
        double r54751 = r54749 - r54750;
        double r54752 = r54748 * r54751;
        return r54752;
}
↓
double f(double a, double b) {
        double r54753 = 1.0;
        double r54754 = b;
        double r54755 = a;
        double r54756 = r54754 - r54755;
        double r54757 = r54753 * r54756;
        double r54758 = atan2(1.0, 0.0);
        double r54759 = 2.0;
        double r54760 = r54758 / r54759;
        double r54761 = r54754 + r54755;
        double r54762 = r54760 / r54761;
        double r54763 = r54762 * r54753;
        double r54764 = r54757 * r54763;
        double r54765 = expm1(r54764);
        double r54766 = log1p(r54765);
        double r54767 = r54755 * r54754;
        double r54768 = r54756 * r54767;
        double r54769 = r54766 / r54768;
        return r54769;
}

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))))