Average Error: 14.7 → 0.3
Time: 4.0s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{b + a} \cdot \pi}{\left(b - a\right) \cdot \frac{2}{\frac{1}{a} - \frac{1}{b}}}\]
\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{1}{b + a} \cdot \pi}{\left(b - a\right) \cdot \frac{2}{\frac{1}{a} - \frac{1}{b}}}
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) (1.0 / ((double) (b + a)))) * ((double) M_PI))) / ((double) (((double) (b - a)) * ((double) (2.0 / ((double) (((double) (1.0 / a)) - ((double) (1.0 / b))))))))));
}

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 14.7

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

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.5

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

  6. Applied frac-times9.5

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

  7. Applied associate-*l/0.3

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

  8. Simplified0.3

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

  10. Applied associate-*r*0.3

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

  11. Simplified0.3

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

  13. Applied associate-/l*0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020184 
(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))))