Average Error: 14.4 → 0.3
Time: 3.4s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a + b} \cdot \left(1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}{\left(b - a\right) \cdot 2}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi}{a + b} \cdot \left(1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}{\left(b - a\right) \cdot 2}
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) M_PI) / ((double) (a + b)))) * ((double) (1.0 * ((double) (((double) (1.0 / a)) - ((double) (1.0 / b)))))))) / ((double) (((double) (b - a)) * 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 14.4

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

  2. Simplified14.4

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

  4. Applied associate-*r/14.4

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

  5. Applied associate-*r/14.4

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

  6. Simplified14.4

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

  8. Applied times-frac14.4

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

  9. Simplified0.3

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

  11. Applied associate-*r/0.3

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

  12. Applied frac-times0.3

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

  13. Simplified0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020181 
(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))))