Average Error: 14.2 → 0.3
Time: 5.4s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{a + b}{\pi}}}{2 \cdot \left(b - a\right)}\]
\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{\frac{1}{a} - \frac{1}{b}}{\frac{a + b}{\pi}}}{2 \cdot \left(b - a\right)}
(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))
↓
(FPCore (a b)
 :precision binary64
 (/ (/ (- (/ 1.0 a) (/ 1.0 b)) (/ (+ a b) PI)) (* 2.0 (- b a))))
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)) / ((a + b) / ((double) M_PI))) / (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 14.2

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

  3. Applied difference-of-squares_binary64_3889.7

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

  4. Applied associate-/r*_binary64_3639.3

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

  6. Applied frac-times_binary64_4299.3

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

  7. Applied associate-*l/_binary64_3620.3

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

  8. Simplified0.3

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

  10. Applied clear-num_binary64_4180.3

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

  11. Simplified0.3

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

  13. Applied un-div-inv_binary64_4170.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020346 
(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))))