Average Error: 14.3 → 0.3
Time: 3.2s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \pi\right)}{a + b}}{2 \cdot \left(b - a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1 \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \pi\right)}{a + b}}{2 \cdot \left(b - a\right)}
double code(double a, double b) {
	return ((double) (((double) ((((double) M_PI) / 2.0) * (1.0 / ((double) (((double) (b * b)) - ((double) (a * a))))))) * ((double) ((1.0 / a) - (1.0 / b)))));
}
↓
double code(double a, double b) {
	return ((((double) (1.0 * ((double) (((double) ((1.0 / a) - (1.0 / b))) * ((double) M_PI))))) / ((double) (a + b))) / ((double) (2.0 * ((double) (b - a)))));
}

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 14.3

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.0

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

  7. Applied associate-*r/8.9

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

  8. Applied frac-times8.9

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

  9. Applied associate-*l/0.3

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

  10. Simplified0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied associate-*r/0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020196 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))