Average Error: 36.7 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)\right)} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)\right)} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r236421 = x;
        double r236422 = eps;
        double r236423 = r236421 + r236422;
        double r236424 = sin(r236423);
        double r236425 = sin(r236421);
        double r236426 = r236424 - r236425;
        return r236426;
}
↓
double f(double x, double eps) {
        double r236427 = x;
        double r236428 = sin(r236427);
        double r236429 = r236428 * r236428;
        double r236430 = r236429 * r236428;
        double r236431 = eps;
        double r236432 = cos(r236431);
        double r236433 = 1.0;
        double r236434 = r236432 - r236433;
        double r236435 = r236434 * r236434;
        double r236436 = r236435 * r236434;
        double r236437 = r236430 * r236436;
        double r236438 = cbrt(r236437);
        double r236439 = cos(r236427);
        double r236440 = sin(r236431);
        double r236441 = r236439 * r236440;
        double r236442 = r236438 + r236441;
        return r236442;
}

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
Target14.8
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-sum21.8

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

  5. Applied *-un-lft-identity21.8

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

  6. Applied *-un-lft-identity21.8

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

  7. Applied distribute-lft-out--21.8

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

  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-cbrt-cube0.4

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

  11. Applied add-cbrt-cube0.4

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

  12. Applied cbrt-unprod0.4

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

  13. Simplified0.4

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

  15. Applied add-cbrt-cube0.4

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

  16. Applied add-cbrt-cube0.5

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

  17. Applied cbrt-unprod0.4

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

  18. Applied rem-cube-cbrt0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2019356 
(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)))