Average Error: 36.9 → 0.4
Time: 6.2s
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 r172538 = x;
        double r172539 = eps;
        double r172540 = r172538 + r172539;
        double r172541 = sin(r172540);
        double r172542 = sin(r172538);
        double r172543 = r172541 - r172542;
        return r172543;
}
↓
double f(double x, double eps) {
        double r172544 = x;
        double r172545 = sin(r172544);
        double r172546 = eps;
        double r172547 = cos(r172546);
        double r172548 = 1.0;
        double r172549 = r172547 - r172548;
        double r172550 = 3.0;
        double r172551 = pow(r172549, r172550);
        double r172552 = cbrt(r172551);
        double r172553 = r172545 * r172552;
        double r172554 = cos(r172544);
        double r172555 = sin(r172546);
        double r172556 = r172554 * r172555;
        double r172557 = r172553 + r172556;
        return r172557;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.7

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

  5. Applied *-un-lft-identity21.7

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

  6. Applied *-un-lft-identity21.7

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

  7. Applied distribute-lft-out--21.7

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

  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 2020057 
(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)))