Average Error: 14.7 → 0.2
Time: 9.4s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot 0.5}{a + b}}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi \cdot 0.5}{a + b}}{a \cdot 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 (/ (/ (* PI 0.5) (+ a b)) (* 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 ((((double) M_PI) * 0.5) / (a + b)) / (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 14.7

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

  2. Simplified14.6

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

  4. Applied difference-of-squares_binary64_7299.8

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

  5. Applied *-un-lft-identity_binary64_7609.8

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

  6. Applied times-frac_binary64_7669.3

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

  7. Applied associate-*l*_binary64_7010.3

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

  8. Simplified0.3

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

  9. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{b + a} \cdot \color{blue}{\left(0.5 \cdot \frac{\pi}{a \cdot b}\right)}\]
  10. Using strategy rm

  11. Applied associate-*r/_binary64_7020.3

    \[\leadsto \frac{1}{b + a} \cdot \color{blue}{\frac{0.5 \cdot \pi}{a \cdot b}}\]

  12. Applied associate-*r/_binary64_7020.3

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2021032 
(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))))