Average Error: 36.8 → 0.4
Time: 26.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r55374 = x;
        double r55375 = eps;
        double r55376 = r55374 + r55375;
        double r55377 = sin(r55376);
        double r55378 = sin(r55374);
        double r55379 = r55377 - r55378;
        return r55379;
}
↓
double f(double x, double eps) {
        double r55380 = x;
        double r55381 = cos(r55380);
        double r55382 = 0.5;
        double r55383 = eps;
        double r55384 = r55382 * r55383;
        double r55385 = cos(r55384);
        double r55386 = r55381 * r55385;
        double r55387 = sin(r55380);
        double r55388 = sin(r55384);
        double r55389 = r55387 * r55388;
        double r55390 = 3.0;
        double r55391 = pow(r55389, r55390);
        double r55392 = cbrt(r55391);
        double r55393 = r55386 - r55392;
        double r55394 = 2.0;
        double r55395 = r55394 * r55388;
        double r55396 = r55393 * r55395;
        return r55396;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
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 36.8

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

  3. Applied diff-sin37.1

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified14.8

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  5. Taylor expanded around inf 14.8

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)}\]

  6. Simplified14.8

    \[\leadsto \color{blue}{\cos \left(x + \frac{1}{2} \cdot \varepsilon\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
  7. Using strategy rm

  8. Applied cos-sum0.3

    \[\leadsto \color{blue}{\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.4

    \[\leadsto \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin x \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  11. Applied add-cbrt-cube0.4

    \[\leadsto \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}} \cdot \sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  12. Applied cbrt-unprod0.4

    \[\leadsto \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right) \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  13. Simplified0.4

    \[\leadsto \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  14. Final simplification0.4

    \[\leadsto \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sqrt[3]{{\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}}\right) \cdot \left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(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)))