Average Error: 14.0 → 0.3
Time: 10.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r52663 = atan2(1.0, 0.0);
        double r52664 = 2.0;
        double r52665 = r52663 / r52664;
        double r52666 = 1.0;
        double r52667 = b;
        double r52668 = r52667 * r52667;
        double r52669 = a;
        double r52670 = r52669 * r52669;
        double r52671 = r52668 - r52670;
        double r52672 = r52666 / r52671;
        double r52673 = r52665 * r52672;
        double r52674 = r52666 / r52669;
        double r52675 = r52666 / r52667;
        double r52676 = r52674 - r52675;
        double r52677 = r52673 * r52676;
        return r52677;
}
↓
double f(double a, double b) {
        double r52678 = atan2(1.0, 0.0);
        double r52679 = 2.0;
        double r52680 = r52678 / r52679;
        double r52681 = b;
        double r52682 = a;
        double r52683 = r52681 + r52682;
        double r52684 = r52680 / r52683;
        double r52685 = 1.0;
        double r52686 = r52684 * r52685;
        double r52687 = r52685 / r52682;
        double r52688 = r52685 / r52681;
        double r52689 = r52687 - r52688;
        double r52690 = r52686 * r52689;
        double r52691 = r52681 - r52682;
        double r52692 = r52690 / r52691;
        return r52692;
}

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 14.0

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

  3. Applied difference-of-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.9

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.8

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))