Average Error: 37.2 → 0.4
Time: 6.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r118232 = x;
        double r118233 = eps;
        double r118234 = r118232 + r118233;
        double r118235 = sin(r118234);
        double r118236 = sin(r118232);
        double r118237 = r118235 - r118236;
        return r118237;
}
↓
double f(double x, double eps) {
        double r118238 = x;
        double r118239 = sin(r118238);
        double r118240 = eps;
        double r118241 = cos(r118240);
        double r118242 = r118239 * r118241;
        double r118243 = -r118239;
        double r118244 = r118242 + r118243;
        double r118245 = cos(r118238);
        double r118246 = sin(r118240);
        double r118247 = r118245 * r118246;
        double r118248 = r118244 + r118247;
        return r118248;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
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 37.2

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

  3. Applied sin-sum22.0

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

  4. Applied associate--l+22.0

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

  5. Taylor expanded around inf 22.0

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

  6. Simplified0.4

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

  8. Applied sub-neg0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020018 
(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)))