Average Error: 36.7 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \log \left(e^{\cos \varepsilon + 1}\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \log \left(e^{\cos \varepsilon + 1}\right) + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r103403 = x;
        double r103404 = eps;
        double r103405 = r103403 + r103404;
        double r103406 = sin(r103405);
        double r103407 = sin(r103403);
        double r103408 = r103406 - r103407;
        return r103408;
}
↓
double f(double x, double eps) {
        double r103409 = x;
        double r103410 = sin(r103409);
        double r103411 = eps;
        double r103412 = cos(r103411);
        double r103413 = 3.0;
        double r103414 = pow(r103412, r103413);
        double r103415 = 1.0;
        double r103416 = r103414 - r103415;
        double r103417 = r103412 + r103415;
        double r103418 = exp(r103417);
        double r103419 = log(r103418);
        double r103420 = r103412 * r103419;
        double r103421 = r103420 + r103415;
        double r103422 = r103416 / r103421;
        double r103423 = r103410 * r103422;
        double r103424 = cos(r103409);
        double r103425 = sin(r103411);
        double r103426 = r103424 * r103425;
        double r103427 = r103423 + r103426;
        return r103427;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
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 36.7

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

  3. Applied sin-sum21.5

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

  4. Taylor expanded around inf 21.5

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

  5. Simplified0.4

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

  7. Applied flip3--0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

    \[\leadsto \sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1}} + \cos x \cdot \sin \varepsilon\]
  10. Using strategy rm

  11. Applied add-log-exp0.4

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

  12. Applied add-log-exp0.4

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

  13. Applied sum-log0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020024 
(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)))