Average Error: 37.6 → 0.4
Time: 11.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r518 = x;
        double r519 = eps;
        double r520 = r518 + r519;
        double r521 = sin(r520);
        double r522 = sin(r518);
        double r523 = r521 - r522;
        return r523;
}
↓
double f(double x, double eps) {
        double r524 = x;
        double r525 = sin(r524);
        double r526 = eps;
        double r527 = cos(r526);
        double r528 = 1.0;
        double r529 = r527 - r528;
        double r530 = 3.0;
        double r531 = pow(r529, r530);
        double r532 = cbrt(r531);
        double r533 = r525 * r532;
        double r534 = cos(r524);
        double r535 = sin(r526);
        double r536 = r534 * r535;
        double r537 = r533 + r536;
        return r537;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied sin-sum22.1

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

  5. Applied *-un-lft-identity22.1

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

  6. Applied *-un-lft-identity22.1

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

  7. Applied distribute-lft-out--22.1

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

  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. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 
(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)))