Average Error: 36.9 → 0.4
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \log \left(e^{\frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1}}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \log \left(e^{\frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1}}\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r95565 = x;
        double r95566 = eps;
        double r95567 = r95565 + r95566;
        double r95568 = sin(r95567);
        double r95569 = sin(r95565);
        double r95570 = r95568 - r95569;
        return r95570;
}
↓
double f(double x, double eps) {
        double r95571 = x;
        double r95572 = sin(r95571);
        double r95573 = eps;
        double r95574 = cos(r95573);
        double r95575 = 3.0;
        double r95576 = pow(r95574, r95575);
        double r95577 = 1.0;
        double r95578 = r95576 - r95577;
        double r95579 = r95574 + r95577;
        double r95580 = r95574 * r95579;
        double r95581 = r95580 + r95577;
        double r95582 = r95578 / r95581;
        double r95583 = exp(r95582);
        double r95584 = log(r95583);
        double r95585 = r95572 * r95584;
        double r95586 = cos(r95571);
        double r95587 = sin(r95573);
        double r95588 = r95586 * r95587;
        double r95589 = r95585 + r95588;
        return r95589;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.7

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

  5. Applied *-un-lft-identity21.7

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

  6. Applied *-un-lft-identity21.7

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

  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. Simplified0.4

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

  10. Applied add-log-exp0.4

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

  11. Applied add-log-exp0.4

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

  12. Applied diff-log0.4

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

  13. Simplified0.4

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

  15. Applied flip3--0.4

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

  16. Simplified0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 
(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)))