2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 6.4s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
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 37.1

   \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm

3. Applied sin-sum 21.6

   \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
4. Using strategy rm

5. Applied add-cube-cbrt 22.2

   \[\leadsto \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}\]

6. Applied add-sqr-sqrt 42.8

   \[\leadsto \color{blue}{\sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon} \cdot \sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}} - \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}\]

7. Applied prod-diff 42.9

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}, \sqrt{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}, -\sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right)}\]

8. Simplified 22.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)} + \mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right)\]

9. Simplified 0.4

   \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right) + \color{blue}{\mathsf{fma}\left(-\sin x, 1, \sin x\right)}\]
10. Using strategy rm

11. Applied add-log-exp 0.4

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

12. Applied add-log-exp 0.4

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

13. Applied diff-log 0.5

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

14. Simplified 0.4

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

15. Taylor expanded around inf 21.6

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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