Average Error: 14.1 → 0.2
Time: 4.8s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{\frac{0.5 \cdot \pi}{b + a}}{b \cdot a} \]
\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{0.5 \cdot \pi}{b + a}}{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 (/ (/ (* 0.5 PI) (+ b a)) (* 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 ((0.5 * ((double) M_PI)) / (b + a)) / (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 14.1

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

  2. 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. Taylor expanded in b around 0 0.3

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

  4. Applied associate-*r/_binary640.3

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

  5. Applied associate-*l/_binary640.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022081 
(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))))