Average Error: 14.2 → 0.3
Time: 13.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r83693 = atan2(1.0, 0.0);
        double r83694 = 2.0;
        double r83695 = r83693 / r83694;
        double r83696 = 1.0;
        double r83697 = b;
        double r83698 = r83697 * r83697;
        double r83699 = a;
        double r83700 = r83699 * r83699;
        double r83701 = r83698 - r83700;
        double r83702 = r83696 / r83701;
        double r83703 = r83695 * r83702;
        double r83704 = r83696 / r83699;
        double r83705 = r83696 / r83697;
        double r83706 = r83704 - r83705;
        double r83707 = r83703 * r83706;
        return r83707;
}
↓
double f(double a, double b) {
        double r83708 = atan2(1.0, 0.0);
        double r83709 = 2.0;
        double r83710 = r83708 / r83709;
        double r83711 = b;
        double r83712 = a;
        double r83713 = r83711 + r83712;
        double r83714 = r83710 / r83713;
        double r83715 = 1.0;
        double r83716 = r83715 / r83712;
        double r83717 = r83715 / r83711;
        double r83718 = r83716 - r83717;
        double r83719 = r83715 * r83718;
        double r83720 = r83711 - r83712;
        double r83721 = r83719 / r83720;
        double r83722 = r83714 * r83721;
        return r83722;
}

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 14.2

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

  3. Applied difference-of-squares9.4

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

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

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.0

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

  11. Applied associate-*l/0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020100 +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))))