Average Error: 36.5 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r166214 = x;
        double r166215 = eps;
        double r166216 = r166214 + r166215;
        double r166217 = sin(r166216);
        double r166218 = sin(r166214);
        double r166219 = r166217 - r166218;
        return r166219;
}
↓
double f(double x, double eps) {
        double r166220 = x;
        double r166221 = sin(r166220);
        double r166222 = eps;
        double r166223 = cos(r166222);
        double r166224 = 1.0;
        double r166225 = r166223 - r166224;
        double r166226 = r166221 * r166225;
        double r166227 = 3.0;
        double r166228 = pow(r166226, r166227);
        double r166229 = cbrt(r166228);
        double r166230 = cos(r166220);
        double r166231 = sin(r166222);
        double r166232 = r166230 * r166231;
        double r166233 = r166229 + r166232;
        return r166233;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.5

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

  3. Applied sin-sum21.3

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

  5. Applied *-un-lft-identity21.3

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

  6. Applied *-un-lft-identity21.3

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

  7. Applied distribute-lft-out--21.3

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

  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 sub-neg0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Simplified0.4

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

  14. Applied add-cbrt-cube0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 
(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)))