Average Error: 14.9 → 0.3
Time: 14.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}\]
\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{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}
double f(double a, double b) {
        double r41789 = atan2(1.0, 0.0);
        double r41790 = 2.0;
        double r41791 = r41789 / r41790;
        double r41792 = 1.0;
        double r41793 = b;
        double r41794 = r41793 * r41793;
        double r41795 = a;
        double r41796 = r41795 * r41795;
        double r41797 = r41794 - r41796;
        double r41798 = r41792 / r41797;
        double r41799 = r41791 * r41798;
        double r41800 = r41792 / r41795;
        double r41801 = r41792 / r41793;
        double r41802 = r41800 - r41801;
        double r41803 = r41799 * r41802;
        return r41803;
}
↓
double f(double a, double b) {
        double r41804 = atan2(1.0, 0.0);
        double r41805 = 2.0;
        double r41806 = r41804 / r41805;
        double r41807 = b;
        double r41808 = a;
        double r41809 = r41807 + r41808;
        double r41810 = r41806 / r41809;
        double r41811 = 1.0;
        double r41812 = r41808 * r41807;
        double r41813 = r41811 / r41812;
        double r41814 = r41810 * r41813;
        return r41814;
}

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 14.9

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

  3. Applied difference-of-squares10.1

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

  4. Applied *-un-lft-identity10.1

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

  5. Applied times-frac9.6

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

  6. Applied associate-*r*9.5

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

  7. Simplified9.5

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

  9. Applied associate-*l*0.3

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

  10. Taylor expanded around 0 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019208 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))