Average Error: 14.7 → 0.3
Time: 5.9s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}}}{b \cdot a}\]
\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}}}}{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))) (* 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 / ((b + a) / (((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 14.7

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

  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 frac-sub_binary64_77614.7

    \[\leadsto \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}}\]

  5. Applied associate-*r/_binary64_70914.7

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

  6. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{2}}{b + a} \cdot 1}}{a \cdot b}\]
  7. Using strategy rm

  8. Applied clear-num_binary64_7660.3

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

  10. Applied associate-/l*_binary64_7120.3

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}}}{\frac{a \cdot b}{1}}}\]

  11. Simplified0.3

    \[\leadsto \frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}}}{\color{blue}{b \cdot a}}\]

  12. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\frac{b + a}{\frac{\pi}{2}}}}{b \cdot a}\]

Reproduce

herbie shell --seed 2020280 
(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))))