2sin (example 3.3)

Average Error: 36.7 → 0.4

Time: 8.6s

Precision: binary64

Math

\[\sin \left(x + \varepsilon\right) - \sin x \]

↓

\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon - 1\right)\right) \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

↓

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (sin x) (- (cos eps) 1.0))))

C

double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}

↓

double code(double x, double eps) {
	return fma(cos(x), sin(eps), (sin(x) * (cos(eps) - 1.0)));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	15.0
Herbie	0.4

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

1. Initial program 36.7

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Applied sin-sum_binary64 21.6

   \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]

3. Applied associate--l+_binary64 21.6

   \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

4. Applied flip-+_binary64 23.6

   \[\leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \cos \varepsilon\right) - \left(\cos x \cdot \sin \varepsilon - \sin x\right) \cdot \left(\cos x \cdot \sin \varepsilon - \sin x\right)}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \]

5. Simplified 5.9

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

6. Simplified 5.8

   \[\leadsto \frac{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}} \]

7. Applied *-un-lft-identity_binary64 5.8

   \[\leadsto \frac{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}} \]

8. Applied times-frac_binary64 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right)}{1} \cdot \frac{\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}} \]

9. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon - 1\right)\right)} \cdot \frac{\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)}{\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, -\cos x, \sin x\right)\right)} \]

10. Simplified 0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \color{blue}{1} \]

11. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon - 1\right)\right) \]

Reproduce

herbie shell --seed 2022103 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))