Average Error: 14.3 → 0.3
Time: 6.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b - a}}{2 \cdot \left(b + a\right)}\]
\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(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b - a}}{2 \cdot \left(b + a\right)}
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 * ((((double) M_PI) / a) - (((double) M_PI) / b))) / (b - a)) / (2.0 * (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-squares9.5

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

  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*8.9

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

  8. Applied frac-times8.9

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

  9. Applied associate-*l/8.9

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

  10. Applied associate-*l/0.3

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

  11. Simplified0.3

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

  13. Applied associate-*r/0.3

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

  14. Taylor expanded around 0 0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020078 +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))))