Average Error: 14.0 → 0.3
Time: 2.4s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\pi}{a} - \frac{\pi}{b}\right) \cdot \frac{0.5}{a + b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{\pi}{a} - \frac{\pi}{b}\right) \cdot \frac{0.5}{a + b}}{b - 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
 (/ (* (- (/ PI a) (/ PI b)) (/ 0.5 (+ a b))) (- 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 (((((double) M_PI) / a) - (((double) M_PI) / b)) * (0.5 / (a + b))) / (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 14.0

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{0.5}{b + a} \cdot \frac{\frac{\pi}{a} + \frac{-\pi}{b}}{b - a}}\]
  3. Using strategy rm

  4. Applied associate-*r/_binary640.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020220 
(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))))