Average Error: 14.4 → 0.3
Time: 2.5s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)}{a + b}
(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
 (/ (* 0.5 (* PI (/ (- (/ 1.0 a) (/ 1.0 b)) (- b a)))) (+ a b)))
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 (0.5 * (((double) M_PI) * (((1.0 / a) - (1.0 / b)) / (b - a)))) / (a + 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 14.4

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a} + \frac{-\pi}{b}}{b - a}}\]
  3. Using strategy rm

  4. Applied associate-*l/_binary640.3

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{\frac{\pi}{a} + \frac{-\pi}{b}}{b - a}}{b + a}}\]

  5. Simplified0.3

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b - a}}}{b + a}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity_binary640.3

    \[\leadsto \frac{0.5 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{1 \cdot \left(b - a\right)}}}{b + a}\]

  8. Applied div-inv_binary640.3

    \[\leadsto \frac{0.5 \cdot \frac{\frac{\pi}{a} - \color{blue}{\pi \cdot \frac{1}{b}}}{1 \cdot \left(b - a\right)}}{b + a}\]

  9. Applied div-inv_binary640.3

    \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\pi \cdot \frac{1}{a}} - \pi \cdot \frac{1}{b}}{1 \cdot \left(b - a\right)}}{b + a}\]

  10. Applied distribute-lft-out--_binary640.3

    \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\pi \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}{1 \cdot \left(b - a\right)}}{b + a}\]

  11. Applied times-frac_binary640.3

    \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\frac{\pi}{1} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)}}{b + a}\]

  12. Simplified0.3

    \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\pi} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)}{b + a}\]

  13. Final simplification0.3

    \[\leadsto \frac{0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)}{a + b}\]

Reproduce

herbie shell --seed 2020224 
(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))))