Average Error: 36.9 → 0.4
Time: 6.1s
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 r118475 = x;
        double r118476 = eps;
        double r118477 = r118475 + r118476;
        double r118478 = sin(r118477);
        double r118479 = sin(r118475);
        double r118480 = r118478 - r118479;
        return r118480;
}
↓
double f(double x, double eps) {
        double r118481 = x;
        double r118482 = sin(r118481);
        double r118483 = eps;
        double r118484 = cos(r118483);
        double r118485 = 3.0;
        double r118486 = pow(r118484, r118485);
        double r118487 = 1.0;
        double r118488 = r118486 - r118487;
        double r118489 = r118484 + r118487;
        double r118490 = r118484 * r118489;
        double r118491 = r118490 + r118487;
        double r118492 = r118488 / r118491;
        double r118493 = exp(r118492);
        double r118494 = log(r118493);
        double r118495 = r118482 * r118494;
        double r118496 = cos(r118481);
        double r118497 = sin(r118483);
        double r118498 = r118496 * r118497;
        double r118499 = r118495 + r118498;
        return r118499;
}

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)))