Average Error: 14.8 → 0.3
Time: 3.9s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{b - a}\right)\]
\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{\pi}{2}}{a + b} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1}{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
 (* (/ (/ PI 2.0) (+ a b)) (* (- (/ 1.0 a) (/ 1.0 b)) (/ 1.0 (- b a)))))
double code(double a, double b) {
	return ((double) (((double) ((((double) M_PI) / 2.0) * (1.0 / ((double) (((double) (b * b)) - ((double) (a * a))))))) * ((double) ((1.0 / a) - (1.0 / b)))));
}
↓
double code(double a, double b) {
	return ((double) (((((double) M_PI) / 2.0) / ((double) (a + b))) * ((double) (((double) ((1.0 / a) - (1.0 / b))) * (1.0 / ((double) (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 Error: 14.8 bits

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

  2. SimplifiedError: 14.8 bits

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

  4. Applied difference-of-squaresError: 9.6 bits

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

  5. Applied *-un-lft-identityError: 9.6 bits

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

  6. Applied times-fracError: 9.2 bits

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

  7. Applied associate-*l*Error: 0.4 bits

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

  8. SimplifiedError: 0.4 bits

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

  10. Applied associate-*r*Error: 0.4 bits

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

  11. SimplifiedError: 0.3 bits

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

  12. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020204 
(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))))