2sin (example 3.3)

Average Error: 36.9 → 0.4

Time: 6.4s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)\]

C

double f(double x, double eps) {
        double r113655 = x;
        double r113656 = eps;
        double r113657 = r113655 + r113656;
        double r113658 = sin(r113657);
        double r113659 = sin(r113655);
        double r113660 = r113658 - r113659;
        return r113660;
}

↓

double f(double x, double eps) {
        double r113661 = x;
        double r113662 = sin(r113661);
        double r113663 = eps;
        double r113664 = cos(r113663);
        double r113665 = 3.0;
        double r113666 = pow(r113664, r113665);
        double r113667 = 1.0;
        double r113668 = -1.0;
        double r113669 = fma(r113666, r113667, r113668);
        double r113670 = r113664 + r113667;
        double r113671 = fma(r113664, r113670, r113667);
        double r113672 = r113669 / r113671;
        double r113673 = exp(r113672);
        double r113674 = log(r113673);
        double r113675 = cos(r113661);
        double r113676 = sin(r113663);
        double r113677 = r113675 * r113676;
        double r113678 = fma(r113662, r113674, r113677);
        return r113678;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
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 36.9

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

3. Applied sin-sum 21.7

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

5. Applied *-un-lft-identity 21.7

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

6. Applied *-un-lft-identity 21.7

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

7. Applied distribute-lft-out-- 21.7

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

8. Simplified 0.4

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

10. Applied add-log-exp 0.4

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

11. Applied add-log-exp 0.4

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

12. Applied diff-log 0.4

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

13. Simplified 0.4

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

15. Applied flip3-- 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \log \left(e^{\frac{\mathsf{fma}\left({\left(\cos \varepsilon\right)}^{3}, 1, -1\right)}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}\right), \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020001 +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)))