Average Error: 36.9 → 0.5
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left(\cos \varepsilon \cdot \cos \varepsilon - 1\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r98207 = x;
        double r98208 = eps;
        double r98209 = r98207 + r98208;
        double r98210 = sin(r98209);
        double r98211 = sin(r98207);
        double r98212 = r98210 - r98211;
        return r98212;
}
↓
double f(double x, double eps) {
        double r98213 = x;
        double r98214 = sin(r98213);
        double r98215 = eps;
        double r98216 = cos(r98215);
        double r98217 = r98216 * r98216;
        double r98218 = 1.0;
        double r98219 = r98217 - r98218;
        double r98220 = r98214 * r98219;
        double r98221 = r98216 + r98218;
        double r98222 = r98220 / r98221;
        double r98223 = cos(r98213);
        double r98224 = sin(r98215);
        double r98225 = r98223 * r98224;
        double r98226 = r98222 + r98225;
        return r98226;
}

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
Target14.8
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-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 flip--0.5

    \[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

  9. Applied associate-*r/0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020081 
(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)))