Average Error: 13.9 → 0.3
Time: 3.7m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2} \cdot 1}{\frac{1}{\sqrt{1}}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{\frac{b + a}{\sqrt{1}}}}{b - a}\]
\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{\frac{\pi}{2} \cdot 1}{\frac{1}{\sqrt{1}}} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{\frac{b + a}{\sqrt{1}}}}{b - a}
double code(double a, double b) {
	return ((double) (((double) (((double) (((double) M_PI) / 2.0)) * ((double) (1.0 / ((double) (((double) (b * b)) - ((double) (a * a)))))))) * ((double) (((double) (1.0 / a)) - ((double) (1.0 / b))))));
}
↓
double code(double a, double b) {
	return ((double) (((double) (((double) (((double) (((double) (((double) M_PI) / 2.0)) * 1.0)) / ((double) (1.0 / ((double) sqrt(1.0)))))) * ((double) (((double) (((double) (1.0 / a)) - ((double) (1.0 / b)))) / ((double) (((double) (b + a)) / ((double) sqrt(1.0)))))))) / ((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 13.9

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

  3. Applied difference-of-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 associate-/r*8.9

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

  5. Applied associate-*r/8.9

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

  6. Applied associate-*l/0.3

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

  8. Applied clear-num0.3

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

  9. Applied un-div-inv0.3

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

  10. Applied associate-*l/0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied *-un-lft-identity0.3

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

  14. Applied times-frac0.3

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

  15. Applied div-inv0.3

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

  16. Applied div-inv0.3

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

  17. Applied distribute-lft-out--0.3

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

  18. Applied associate-*r*0.3

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

  19. Applied times-frac0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020114 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))