NMSE Section 6.1 mentioned, B

Average Error: 14.7 → 0.3

Time: 6.8s

Precision: binary64

Math

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

↓

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

FPCore

(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 (/ (fma PI (/ -1.0 b) (/ PI a)) (- b a))) (+ b a)))

C

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 * (fma(((double) M_PI), (-1.0 / b), (((double) M_PI) / a)) / (b - a))) / (b + a);
}

Error

Bits error versus a

Bits error versus b

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. Simplified 0.3

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

3. Applied associate-*l/_binary64 0.3

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

4. Final simplification 0.3

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

Reproduce

herbie shell --seed 2022024 
(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))))