Average Error: 14.1 → 0.3
Time: 6.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{\pi}{2}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{\pi}{2}}}
double code(double a, double b) {
	return (((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b)));
}
↓
double code(double a, double b) {
	return ((((1.0 / a) - (1.0 / b)) * (1.0 / (b + a))) / ((b - a) / (((double) M_PI) / 2.0)));
}

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 14.1

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

  3. Applied difference-of-squares9.3

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

  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 associate-*l*0.3

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

  9. Simplified0.3

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

  11. Applied *-commutative0.3

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

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020071 +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))))