Average Error: 14.2 → 0.2
Time: 6.2s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{a + b} \]
\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 \cdot b}}{a + 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 14.2

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

  2. Simplified0.3

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

  3. Applied associate-*l/_binary640.3

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

  4. Simplified0.3

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

  5. Applied div-inv_binary640.3

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

  6. Applied div-inv_binary640.3

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

  7. Applied distribute-lft-out--_binary640.3

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

  8. Applied associate-/l*_binary640.3

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

  9. Applied frac-sub_binary640.3

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

  10. Applied associate-/r/_binary640.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2022097 
(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))))