Average Error: 14.1 → 0.2
Time: 6.2s
Precision: binary64
Cost: 704
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\pi}{a + b}}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\pi}{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 (/ (* 0.5 (/ PI (+ 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 (0.5 * (((double) M_PI) / (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 14.1

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

  2. Simplified14.1

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

  4. Applied difference-of-squares_binary64_7069.1

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

  5. Applied div-inv_binary64_7349.1

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

  6. Applied times-frac_binary64_7438.7

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

  7. Applied associate-*l*_binary64_6780.3

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

  8. Simplified0.3

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

  10. Applied frac-sub_binary64_7460.4

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

  11. Applied associate-*l/_binary64_6800.3

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

  12. Applied associate-*r/_binary64_6790.3

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2021042 
(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))))