Average Error: 14.4 → 0.3
Time: 1.1m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\frac{1 \cdot \pi}{a}}{b}}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\frac{1 \cdot \pi}{a}}{b}}{a + b}
double f(double a, double b) {
        double r2877989 = atan2(1.0, 0.0);
        double r2877990 = 2.0;
        double r2877991 = r2877989 / r2877990;
        double r2877992 = 1.0;
        double r2877993 = b;
        double r2877994 = r2877993 * r2877993;
        double r2877995 = a;
        double r2877996 = r2877995 * r2877995;
        double r2877997 = r2877994 - r2877996;
        double r2877998 = r2877992 / r2877997;
        double r2877999 = r2877991 * r2877998;
        double r2878000 = r2877992 / r2877995;
        double r2878001 = r2877992 / r2877993;
        double r2878002 = r2878000 - r2878001;
        double r2878003 = r2877999 * r2878002;
        return r2878003;
}
↓
double f(double a, double b) {
        double r2878004 = 0.5;
        double r2878005 = 1.0;
        double r2878006 = atan2(1.0, 0.0);
        double r2878007 = r2878005 * r2878006;
        double r2878008 = a;
        double r2878009 = r2878007 / r2878008;
        double r2878010 = b;
        double r2878011 = r2878009 / r2878010;
        double r2878012 = r2878004 * r2878011;
        double r2878013 = r2878008 + r2878010;
        double r2878014 = r2878012 / r2878013;
        return r2878014;
}

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 14.4

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

  3. Applied difference-of-squares9.5

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

  4. Applied add-sqr-sqrt9.5

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  8. Applied associate-*r/9.0

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

  9. Applied associate-*l/9.0

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{\left(\sqrt{1}\right)}^{2} \cdot \pi}{a \cdot b}}}{b + a}\]

  12. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot \pi}{a \cdot b} \cdot 0.5}}{b + a}\]
  13. Using strategy rm

  14. Applied associate-/r*0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 \cdot \pi}{a}}{b}} \cdot 0.5}{b + a}\]

  15. Final simplification0.3

    \[\leadsto \frac{0.5 \cdot \frac{\frac{1 \cdot \pi}{a}}{b}}{a + b}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))