Average Error: 14.5 → 0.3
Time: 3.7s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot \left(\frac{1}{b - a} \cdot \left(1 \cdot \left(b - a\right)\right)\right)}{b \cdot a}\]
\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{1}{\frac{b + a}{\frac{\pi}{2}}} \cdot \left(\frac{1}{b - a} \cdot \left(1 \cdot \left(b - a\right)\right)\right)}{b \cdot a}
(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 (/ (+ b a) (/ PI 2.0))) (* (/ 1.0 (- b a)) (* 1.0 (- b a))))
  (* 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) ((1.0 / (((double) (b + a)) / (((double) M_PI) / 2.0))) * ((double) ((1.0 / ((double) (b - a))) * ((double) (1.0 * ((double) (b - a)))))))) / ((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: 14.5 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. Using strategy rm

  3. Applied difference-of-squaresError: 9.5 bits

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

  4. Applied *-un-lft-identityError: 9.5 bits

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

  5. Applied times-fracError: 9.0 bits

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

  6. Applied associate-*r*Error: 8.9 bits

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

  7. SimplifiedError: 8.9 bits

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

  9. Applied frac-subError: 8.9 bits

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

  10. Applied associate-*r/Error: 8.9 bits

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

  11. SimplifiedError: 0.3 bits

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

  13. Applied clear-numError: 0.3 bits

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

  14. SimplifiedError: 0.3 bits

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

  15. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020203 
(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))))